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Full text of "Shore protection manual"

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U.S. Army 
Coastal Engineering 
Research Centei 



DATA LIBRARY 

Woods Hole Oceanographic Institution 



SHORE PROTECTION 
MANUAL 




1/. 1 



DEPARTMENT OF THE ARMY 

CORPS OF ENGINEERS 

1975 



Reprint or republication of any of this material shall give appropriate 
credit to the U.S. Army Coastal Engineering Research Center. 

U.S. Army Coastal Engineering Research Center 

Kingman Building 

Fort Belvoir, Virginia 22060 



SONOMA COAST, CALIFORNIA (GOAT ROCK) - 10 December 1958 



SHORE PROTECTION 
MANUAL 



DATA LIBRARY 

Woods Hole Oceanographic Institution 




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VOLUME I 

( Chapters 1 Through 4 



S° U.S. ARMY COASTAL ENGINEERING RESEARCH CENTER 



1975 

Second Edition 



For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washintrton, D.C. 20402 - Price $15.05 per 3-part set. (sold in sets only) 

Stock Number 008-022-00077-1 Catalog Number D 103.42/6:SH7/V.l-3 



PREFACE 

The U.S. Army Coastal Engineering Research Center, (CERC) formerly the Beach Erosion Board, has, 
since 1930, conducted studies on shore processes and methods of shore protection. CERC continues an 
extensive research and development program to improve both shore protection and offsliore engineering 
techniques. The scientific and engineering aspects of coastal processes and offshore structures are in the 
developmental stage and the requirement for improved techniques for use in design and engineering of 
coastal structures is evident. This need was met in 1954, to the extent of available knowledge, by 
publication of Teclmical Report Number 4, "Shore Protection, Planning and Design" (TR 4); revised 
editions thereof appeared in 1957, 1961, and 1966. 

Significant advances in knowledge and capability have been made since the 1966 revision. This Shore 
Protection Manual (SPM) incorporates new material with appropriate information extracted from TR 4, 
and expands coverage within the shore protection field. This SPM is a replacement volume covering 
guidelines and techniques for functional and structural design for shore protection works. Accordingly, 
further editions of TR 4 are not planned. 

The Shore Protection Manual is in three volumes. Volume 1 describes the physical environment in the 
coastal zone starting with an introduction of coastal engineering, continuing with discussions of mechanics 
of wave motion, wave and water level predicitons, and finally littoral processes. 

Volume II translates the interaction of the physical environment and coastal structures into design 
parameters for use in the solution of coastal engineering problems. It discusses planning, analysis, structural 
features, and structural design as related to physical factors, and shows an example of a coastal engineering 
problem which utilizes the technical content of material presented in all three volumes. 

Volume III contains four appendixes including a glossary of coastal engineering terms, a list of symbols, 
tables and plates, and a subject index. 

R. A. Jachowski, Chief, Design Branch, Engineering Development Division, was the project engineer 
responsible for preparation and assemblage of the text, under the general supervision of G. M. Watts, Chief, 
Engineering Development Division. At the time of approval for publication by the Coastal Engineering 
Research Board, LTC Don S. McCoy was Commander and Director and Thorndike Saville, Jr. was Technical 
Director. Members of the Coastal Engineering Research Board were: MG John W. Morris (President), 
MG Daniel A. Raymond, MG Ernest Graves, Jr., BG George B. Fink, Dean Morrough P. O'Brien, Dr. Arthur 
T. Ippen, and Prof. Robert G. Dean. The board members were intimately involved in both the planning and 
review of this manual. 

Preparation of this manual includes the contribution, review and suggestions of numerous engineers, 
scientists, technical and support personnel. Present members of the CERC staff who made significant 
technical contributions to this manual are: R. H. Allen, B. R. Bodine, M. T. Czemiak, A. E. DeWall, D. B. 
Duane, C. J. Galvin, R. J. Hallermeier, D. L. Harris, R. A. Jachowski, W. R. James, 0. M. Madsen, P. C. 
Pritchett, A. C. Rayner, R. L. Rector, R. P. Savage, T. Saville, Jr., P. N. Stoa, P. G. Teleki, G. M. Watts, J. 
R. Weggel and D. W. Woodard. Technical editor for this manuscript was R. H. Allen. Typing and composing 
were done by M. L. Vrooman and C. M. Lowe; and drafting by H. J. Bruder and J. S. Rivas. LCDR K. E. 
Fusch was responsible for the completion of the final manuscript. 

The manual format and binding were selected to optimize its use by scientists and engineers as a learning 
text as well as a field and office engineering reference. Chapters include a bibliography. The binding 
facilitates text and chart removal for separate use or rebinding in loose leaf form. 

Comments or suggestions on material in this manual are invited. 

This report is published under authority of Public Law 166, 79'^ Congress, approved July 31, 1945, as 
supplemented by Public Law 172, 88''' Congress, approved November 7, 1963. 



Ill 



CHAPTER 
5 

5.1 
5.2 
5.3 
5.4 
5.5 
5.6 
5.7 
5.8 
5.9 
5.10 

6 

6.1 

6.2 

6.3 

6.4 

6.5 

6.6 

6.7 

6.8 

6.9 

6.10 

6.11 

7 

7.1 

7.2 

7.3 

7.4 

7.5 

7.6 

7.7 



8 

8.1 
8.2 



APPENDIX 
A 
B 
C 
D 



TABLE OF CONTENTS 
VOLUME n 

PAGE 

PLANNING ANALYSIS 

GENERAL 5-1 

SEAWALLS. BULKHEADS, AND REVETMETS 5-3 

PROTECTIVE BEACHES . ^ 5-7 

SAND DUNES 5-21 

SAND BYPASSING 5-24 

GROINS 5-31 

JETTIES 5-46 

BREAKWATERS-SHORE-CONNECTED 5-49 

BREAKWATERS-OFFSHORE 5-50 

ENVIRONMENTAL CONSIDERATIONS 5-57 

REFERENCES AND SELECTED BIBLIOGRAPHY 5-58 

STRUCTURAL FEATURES 

INTRODUCTION 6-1 

SEAWALLS, BULKHEADS, AND REVETMENTS 6-1 

PROTECTIVE BEACHES 6-16 

SAND DUNES 6-36 

SAND BYPASSING 6-54 

GROINS 6-76 

JETTIES 6-84 

BREAKWATERS-SHORE-CONNECTED 6-88 

BREAKWATERS-OFFSHORE 6-96 

CONSTRUCTION MATERIALS 6-96 

MISCELLANEOUS DESIGN PRACTICES 6-98 

REFERENCES AND SELECTED BIBLIOGRAPHY 6-101 

STRUCTURAL DESIGN-PHYSICAL FACTORS 

WAVE CHARACTERISTICS 7-1 

WAVE RUNUP, OVERTOPPING, AND TRANSMISSION 7-15 

WAVE FORCES 7-63 

VELOCITY FORCES-STABILITY OF CHANNEL REVETMENTS 7-203 

IMPACT FORCES 7-204 

ICE FORCES 7-206 

EARTH FORCES 7-208 

REFERENCES AND SELECTED BIBLIOGRAPHY 7-214 

ENGINEERING ANALYSIS-CASE STUDY 

INTRODUCTION 8-1 

DESIGN PROBLEM CLACULATIONS-ARTIFICIAL OFFSHORE ISLAND . . 8-2 

REFERENCES 8-132 

VOLUME III 

GLOSSARY OF TERMS A-1 

LIST OF SYMBOLS B-1 

MISCELLANEOUS TABLES AND PLATES C-1 

SUBJECT INDEX D-1 



IV 



TABLE OF CONTENTS 
VOLUME I 
CHAPTER 1 - INTRODUCTION TO COASTAL ENGINEERING 

SECTION PAGE 

1.1 INTRODUCTION TO THE SHORE PROTECTION MANUAL 1-1 

1.2 THE SHORE ZONE 1-2 

1.21 NATURAL BEACH PROTECTION 1-2 

1.22 NATURAL PROTECTIVE DUNES 1-2 

1.23 BARRIER BEACHES, LAGOONS AND INLETS 1-5 

1.24 STORM ATTACK 1-5 

1.25 ORIGIN AND MOVEMENT OF BEACH SANDS 1-5 

1.3 THE SEA IN MOTION 1-5 

1.31 TIDES AND WINDS 1-5 

1.32 WAVES 1-5 

1.33 CURRENTS AND SURGES 1-7 

1.34 TIDAL CURRENTS 1-9 

1.4 THE BEHAVIOR OF BEACHES 1-9 

1.41 BEACH COMPOSITION 1-9 

1.42 BEACH CHARACTERISTICS 1-9 

1.43 BREAKERS 1-10 

1.44 EFFECTS OF WIND WAVES 1-10 

1.45 LITTORAL TRANSPORT 1-10 

1.46 EFFECT OF INLETS ON BARRIER BEACHES 1-12 

1.47 IMPACT OF STORMS 1-12 

1.48 BEACH STABILITY 1-13 

1.5 EFFECTS OF MAN ON THE SHORE 1-13 

1.51 ENCROACHMENT ON THE SEA 1-13 

1.52 NATURAL PROTECTION 1-14 

1.53 SHORE PROTECTION METHODS 1-14 

1.54 BULKHEADS, SEAWALLS AND REVETMENTS 1-16 

1.55 BREAKWATERS 1-16 

1.56 GROINS 1-17 

1.57 JETTIES 1-17 

1.58 BEACH RESTORATION AND NOURISHMENT 1-18 

1.6 CONSERVATION OF SAND 1-21 

CHAPTER 2 - MECHANICS OF WAVE MOTION 

2.1 INTRODUCTION 2-1 

2.2 WAVE MECHANICS 2-1 

2,21 GENERAL 2-1 



SECTION PAGE 

2.22 WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES 2-3 

2.23 ELEMENTARY PROGRESSIVE WAVE THEORY 

(Small-Amplitude Wave Theory) 2-7 

2.231 Wave Celerity, Length and Period 2-7 

2.232 The Sinusoidal Wave Profile 2-10 

2.233 Some Useful Functions 2-10 

2.234 Local Fluid Velocities and Accelerations 2-12 

2.235 Water Particle Displacements 2-15 

2.236 Subsurface Pressure 2-22 

2.237 Velocity of a Wave Group 2-24 

2.238 Wave Energy and Power 2-27 

2.239 Summary - Linear Wave Theory 2-33 

2.24 HIGHER ORDER WAVE THEORIES 2-33 

2.25 STOKES' PROGRESSIVE, SECOND-ORDER WAVE THEORY 2-36 

2.251 Wave Celerity, Length and Surface Profile 2-37 

2.252 Water Particle Velocities and Displacements 2-37 

2.253 Mass Transport Velocity 2-38 

2.254 Subsurface Pressure 2-39 

2.255 Maximum Steepness of Progressive Waves 2-39 

2.256 Comparison of the First- and Second-Order Theories . . 2-39 

2.26 CNOIDAL WAVES 2-47 

2.27 SOLITARY WAVE THEORY 2-59 

2.28 STREAM FUNCTION WAVE THEORY 2-62 

2.3 WAVE REFRACTION 2-62 

2.31 INTRODUCTION 2-62 

2.32 GENERAL - REFRACTION BY BATHYMETRY 2-65 

2.321 Procedures in Refraction Diagram Construction - 

Orthogonal Method 2-69 

2.322 Procedure when a is Less than 80 Degrees 2-71 

2.323 Procedure when a is Greater than 80 Degrees - The 

R/J Method 2-73 

2.324 Refraction Fan Diagrams 2-73 

2.325 Other Graphical Methods of Refraction Analysis .... 2-75 

2.326 Computer Methods for Refraction Analysis 2-75 

2.327 Interpretation of Results and Diagram Limitations. . . 2-75 

2.328 Refraction of Ocean Waves 2-78 

2.4 WAVE DIFFRACTION 2-79 

2.41 INTRODUCTION 2-79 

2.42 DIFFRACTION CALCULATIONS 2-81 

2.421 Waves Passing a Single Breakwater 2-81 

2.422 Waves Passing a Gap of Width Less than Five 

Wavelengths at Normal Incidence 2-98 

2.423 Waves Passing a Gap of Width Greater than Five 

Wavelengths at Normal Incidence 2-98 

2.424 Diffraction at a Gap-Oblique Incidence 2-98 

2.43 REFRACTION AND DIFFRACTION COMBINED 2-98 



VI 



SECTION PAGE 

2.5 WAVE REFLECTION 2-110 

2.51 GENERAL 2-110 

2.52 REFLECTION FROM IMPERMEABLE, VERTICAL WALLS 

(Linear Theory) 2-113 

2.53 REFLECTIONS IN AN ENCLOSED BASIN 2-115 

2.54 WAVE REFLECTION FROM BEACHES 2-117 

2.6 BREAKING WAVES 2-120 

2.61 DEEP WATER 2-120 

2.62 SHOALING WATER 2-121 

REFERENCES AND SELECTED BIBLIOGRAPHY 2-129 

CHAPTER 3 - WAVE AND WATER LEVEL PREDICTIONS 

3.1 INTRODUCTION 3-1 

3.2 CHARACTERISTICS OF OCEAN WAVES 3-2 

3.21 SIGNIFICANT WAVE HEIGHT AND PERIOD 3-2 

3.22 WAVE HEIGHT VARIABILITY 3-5 

3.23 ENERGY SPECTRA OF WAVES 3-11 

3.24 DIRECTIONAL SPECTRA OF WAVES 3-13 

3.3 WAVE FIELD 3-15 

3.31 DEVELOPMENT OF A WAVE FIELD 3-15 

3.32 VERIFICATION OF WAVE HINDCASTING 3-17 

3.33 DECAY OF A WAVE FIELD 3-17 

3.4 WIND INFORMATION NEEDED FOR WAVE PREDICTION 3-20 

3.41 ESTIMATING THE WIND CHARACTERISTICS 3-22 

3.42 DELINEATING A FETCH 3-27 

3.43 FORECASTS FOR LAKES, BAYS, AND ESTUARIES 3-29 

3.431 Wind Data 3-29 

3.432 Effective Fetch 3-29 

3.5 SIMPLIFIED WAVE-PREDICTION MODELS 3-33 

3.51 SMB METHOD FOR PREDICTING WAVES IN DEEP WATER 3-35 

3.52 EFFECTS OF MOVING STORMS AND A VARIABLE WIND SPEED 

AND DIRECTION 3-40 

3.53 VERIFICATION OF SIMPLIFIED WAVE HINDCAST PROCEDURES. . . 3-40 

3.54 ESTIMATING WAVE DECAY IN DEEP WATER 3-42 

3.6 WAVE FORECASTING FOR SHALLOW WATER 3-42 

3.61 FORECASTING CURVES 3-42 

3.62 DECAY IN LAKES, BAYS, AND ESTUARIES 3-52 

3.7 HURRICANE WAVES 3-52 

3.71 DESCRIPTION OF HURRICANE WAVES 3-52 

3.72 MODEL WIND AND PRESSURE FIELDS FOR HURRICANES 3-54 

3.73 PREDICTION TECHNIQUE 3-57 

vJi 



SECTION PAGE 

3.8 WATER LEVEL FLUCTUATIONS 3-69 

3.81 ASTRONOMICAL TIDES 3-71 

3.82 TSUNAMIS 3-71 

3.83 LAKE LEVELS 3-76 

3.84 SEICHES 3-78 

3.85 WAVE SETUP 3-80 

3.86 STORM SURGE AND WIND SETUP 3-82 

3.861 General 3-82 

3.862 Storms 3-83 

3.863 Factors of Storm Surge Generation 3-84 

3.864 Initial Water Level 3-85 

3.865 Storm Surge Prediction 3-85 

REFERENCES AND SELECTED BIBLIOGRAPHY 3-145 

CHAPTER 4 - LITTORAL PROCESSES 

4.1 INTRODUCTION 4-1 

4.11 DEFINITIONS 4-1 

4.111 Beach Profile 4-1 

4.112 Areal View 4-1 

4.12 ENVIRONMENTAL FACTORS 4-1 

4.121 Waves 4-1 

4.122 Currents 4-4 

4.123 Tides and Surges 4-5 

4.124 Winds 4-5 

4.125 Geologic Factors 4-6 

4.126 Other Factors 4-6 

4.13 CHANGES IN THE LITTORAL ZONE 4-6 

4.2 LITTORAL MATERIALS 4-11 

4.21 CLASSIFICATION 4-11 

4.211 Size and Size Parameters 4-11 

4.212 Composition 4-18 

4.213 Other Characterisitcs 4-18 

4.22 SAND AND GRAVEL 4-18 

4.23 COHESIVE MATERIALS 4-20 

4.24 CONSOLIDATED MATERIAL 4-21 

4.241 Rock 4-21 

4.242 Beach Rock 4-21 

4.243 Organic Reefs 4-21 

4.25 OCCURRENCE OF LITTORAL MATERIALS ON U.S. COASTS 4-22 

4.251 Atlantic Coast 4-22 

4.252 Gulf Coast 4-22 

4.253 Pacific Coast 4-24 

4.254 Alaska 4-24 

4.255 Hawaii 4-24 

4.256 Great Lakes 4-24 

4.26 SAMPLING LITTORAL MATERIALS 4-24 



VIII 



PAGE 

SIZE ANALYSES 4-26 

Sieve Analysis 4-26 

Settling Tube 4-26 

LITTORAL WAVE CONDITIONS 4-27 

EFFECT OF WAVE CONDITIONS ON SEDIMENT TRANSPORT 4-27 

FACTORS DETERMINING LITTORAL WAVE CLIMATE 4-28 

Offshore Wave Climate 4-28 

Effect of Bottom Topography 4-28 

Winds and Storms 4-29 

INSHORE WA'/E CLIMATE 4-29 

Mean Value Data on U.S. Littoral Wave Climates .... 4-29 

Mean vs. Extreme Conditions 4-30 

OFFICE STUDY OF WAVE CLIMATE 4-35 

EFFECT OF EXTREME EVENTS 4-37 

NEARSHORE CURRENTS 4-39 

WAVE- INDUCED WATER MOTION , 4-39 

FLUID MOTION IN BREAKING WAVES 4-42 

ONSHORE-OFFSHORE CURRENTS 4-43 

Onshore-Offshore Exchange 4-43 

Diffuse Return Flow 4-45 

Rip Currents 4-45 

LONGSHORE CURRENTS 4-45 

Velocity and Flow Rate 4-45 

Velocity Prediction 4-48 

SUMMARY 4-50 

LITTORAL TRANSPORT 4-50 

INTRODUCTION 4-50 

Importance of Littoral Transport 4-50 

Zones of Transport 4-53 

Profiles 4-54 

Profile Accuracy 4-57 

ONSHORE- OFFSHORE TRANSPORT 4-60 

Sediment Effects 4-60 

Initiation of Sediment Motion 4-61 

Seaward Limit of Significant Transport 4-65 

Beach Erosion and Recovery 4-70 

Bar-Berm Prediction 4-78 

Slope of the Foreshore 4-85 

LONGSHORE TRANSPORT RATE 4-88 

Definitions and Methods 4-88 

Energy Flux Method 4-89 

Energy Flux Example (Method 3) 4-102 

Empirical Prediction of Gross Longshore Transport 

Rate (Method 4) 4-107 

4.535 Method 4 Example 4-108 



ix 



SECTION 


4. 


27 


4. 


271 


4. 


272 


4. 


3 


4. 


31 


4. 


32 


4. 


321 


4. 


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4. 


,323 


4. 


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4. 


331 


4. 


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4. 


34 


4. 


35 


4. 


,4 


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.5 


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4, 


.511 


4, 


.512 


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.513 


4, 


.514 


4 


.52 


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4 


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4 


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4 


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.53 


4 


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4 


.532 


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.533 


4 


.534 



SECTION PAGE 

4.6 ROLE OF FOREDUNES IN SHORE PROCESSES 4-111 

4.61 BACKGROUND 4-111 

4.62 ROLE OF FOREDUNES 4-113 

4.621 Prevention of Overtopping 4-113 

4.622 Reservoir of Beach Sand 4-115 

4.623 Long-Term Effects 4-115 

4.7 SEDIMENT BUDGET 4-116 

4.71 DEFINITIONS 4-116 

4.711 Sediment Budget 4-116 

4.712 Elements of Sediment Budget 4-117 

4.713 Sediment Budget Boundaries 4-119 

4.72 SOURCE OF LITTORAL MATERIAL 4-119 

4.721 Rivers 4-119 

4.722 Erosion of Shores and Cliffs 4-121 

4.723 Transport from Offshore Slope 4-121 

4.724 Windblown Sediment Sources 4-123 

4.725 Carbonate Production 4-123 

4.726 Beach Replenishment 4-123 

4.73 SINKS FOR LITTORAL MATERIALS 4-124 

4.731 Inlets and Lagoons 4-124 

4.732 Overwash 4-124 

4.733 Backshore and Dune Storage 4-124 

4.734 Offshore Slopes 4-127 

4.735 Submarine Canyons 4-127 

4.736 Deflation 4-129 

4.737 Carbonate Loss 4-129 

4.738 Mining and Dredging 4-129 

4.74 CONVECTION OF LITTORAL MATERIALS 4-131 

4.75 RELATIVE CHANGE IN SEA LEVEL 4-131 

4.76 SUMMARY OF SEDIMENT BUDGET 4-131 

4.8 ENGINEERING STUDY OF LITTORAL PROCESSES 4-139 

4.81 OFFICE STUDY 4-139 

4.811 Sources of Data 4-139 

4.812 Interpretation of Shoreline Position 4-140 

4.82 FIELD STUDY 4-147 

4.821 Wave Data Collection 4-147 

4.822 Sediment Sampling 4-147 

4.823 Surveys 4-147 

4.824 Tracers 4-150 

4.83 SEDIMENT TRANSPORT CALCULATIONS 4-152 

4.831 Longshore Transport Rate 4-152 

4.832 Onshore-Offshore Motion 4-153 

4.833 Sediment Budget 4-154 

REFERENCES AND SELECTED BIBLIOGRAPHY 4-155 



LIST OF FIGURES 

FIGURE PAGE 

1-1 Beach Profile - Related Terms 1-3 

1-2 Sand Dunes Along the South Shore of Lake Michigan 1-4 

1-3 Sand Dunes, Honeyman State Park, Oregon 1-4 

1-4 Barrier Beach Island Developed as Recreation Park - Jones 

Beach State Park, Long Island, New York 1-6 

},-5 Large Waves Breaking Over a Breakwater 1-8 

1-6 Wave Characteristics 1-8 

1-7 Schematic Diagram of Storm Wave Attack on Beach and Dune . 1-11 

1-8 Backshore Damage at Sea Isle City, New Jersey 1-15 

1-9 Weir Jetty at Masonboro Inlet, North Carolina 1-19 

1-10 Wrightsville Beach, North Carolina, after Completion of 

Beach Restoration and Hurricane Protection Project . , . 1-20 

2-1 Approximate Distribution of Ocean Surface Wave Energy 

Illustrating the Classification of Surface Waves by 

Wave Band, Primary Disturbing Force and Primary 

Restoring Force 2-5 

2-2 Definition of Terms - Elementary, Sinusoidal, Progressive 

Wave 2-8 

2-3 Local Fluid Velocities and Accelerations 2-14 

2-4 Water Particle Displacements from Mean Position for 

Shallow- Water and Deepwater Waves 2-17 

2-5 Formation of Wave Groups by the Addition of Two Sinusoids 

Having Different Periods 2-26 

2-6 Summary - Linear (Airy) Wave Theory - Wave 

Characteristics 2-34 

2-7 Regions of Validity for Various Wave Theories 2-35 

2-8 Comparison of Second-Order Stokes' Profile with Linear 

Profile 2-42 

2-9 Cnoidal Wave Surface Profiles as a Function of k^ 2-50 

2-10 Cnoidal Wave Surface Profiles as a Func tion of k^ 2-51 

2-11 Relationship Between k^, H/d and T/g/d 2-52 

2-12 Relationship Between k^ and L^H/d^ 2-53 

2-13 Relationships Between k^ and L^H/d^ and Between (y^-d)/H, 

(y^ -d)/H)+ 1 and L^H/d^ 2-54 

2-14 Relationship Between T/g/d y^/d, H/y^ and L^H/d^ 2-55 

2-15 Relationship Between C/>/gy^, H/y^ and L^H/d^ 2-56 

2-16 Functions M and N in Solitary Wave Theory 2-61 

2-17 Wave Refraction at West Hampton Beach, Long Island, 

New York 2-64 

2-18 Refraction Template 2-68 

2-19 Changes in Wave Direction and Height Due to Refraction on 

Slopes with Straight, Parallel Depth Contours 2-70 

2-20 Use of the Refraction Template 2-72 

2-21 Refraction Diagram Using R/J Method 2-74 

2-22 Use of Fan-Type Refraction Diagram 2-76 

2-23 Refraction Along a Straight Beach with Parallel Bottom 

Contours 2-77 



XI 



FIGURE PAGE 

2-24 Refraction by a Submarine Ridge and Submarine Canyon . . . 2-77 

2-25 Refraction Along an Irregular Shoreline 2-77 

2-26 Wave Incident on a Breakwater 2-80 

2-27 Wave Diffraction at Channel Islands Harbor Breakwater, 

California 2-82 

2-28 Wave Diffraction Diagram - 15° Wave Angle 2-83 

2-29 Wave Diffraction Diagram - 30° Wave Angle 2-84 

2-30 Wave Diffraction Diagram - 45° Wave Angle 2-85 

2-31 Wave Diffraction Diagram - 60° Wave Angle 2-86 

2-32 Wave Diffraction Diagram - 75° Wave Angle 2-87 

2-33 Wave Diffraction Diagram - 90° Wave Angle 2-88 

2-34 Wave Diffraction Diagram - 105° Wave Angle 2-89 

2-35 Wave Diffraction Diagram - 120° Wave Angle 2-90 

2-36 Wave Diffraction Diagram - 135° Wave Angle 2-91 

2-37 Wave Diffraction Diagram - 150° Wave Angle 2-92 

2-38 Wave Diffraction Diagram - 165° Wave Angle 2-93 

2-39 Wave Diffraction Diagram - 180° Wave Angle 2-94 

2-40 Diffraction for a Single Breakwater Normal Incidence . . . 2-95 

2-41 Schematic Representation of Wave Diffraction Overlay . . . 2-97 
2-42 Generalized Diffraction Diagram for a Breakwater Gap 

Width of Two Wave Lengths (B/L = 2) 2-99 

2-43 Contours of Equal Diffraction Coefficient; Gap Width = 

0.5 Wave Length (B/L =0.5) 2-100 

2-44 Contours of Equal Diffraction Coefficient; Gap Width = 

1 Wave Length (B/L = 1) 2-100 

2-45 Contours of Equal Diffraction Coefficient; Gap Width = 

1.41 Wave Lengths (B/L = 1.41) 2-101 

2-46 Contours of Equal Diffraction Coefficient; Gap Width = 

1.64 Wave Lengths (B/L =1.64) 2-101 

2-47 Contours of Equal Diffraction Coefficient; Gap Width = 

1.78 Wave Lengths (B/L =1.78) 2-102 

2-48 Contours of Equal Diffraction Coefficient; Gap Width = 

2 Wave Lengths (B/L =2) 2-102 

2-49 Contours of Equal Diffraction Coefficient; Gap Width = 

2.50 Wave Lenghts (B/L =2.50) 2-103 

2-50 Contours of Equal Diffraction Coefficient; Gap Width = 

2.95 Wave Lengths (B/L = 2,95) 2-103 

2-51 Contours of Equal Diffraction Coefficient; Gap Width = 

3.82 Wave Lengths (B/L = 3.82) 2-104 

2-52 Contours of Equal Diffraction Coefficient; Gap Width = 

5 Wave Lengths (B/L =5) 2-104 

2-53 Diffraction for a Breakwater Gap of Width > 5L (B/L > 5) . 2-105 

2-54 Wave Incidence Oblique to Breakwater Gap 2-105 

2-55 Diffraction for a Breakwater Gap of One Wave Length Width 

((J) = and 15°) 2-106 

2-56 Diffraction for a Breakwater Gap of One Wave Length Width 

((fi = 30 and 45°) 2-107 

2-57 Diffraction for a Breakwater Gap of One Wave Length Width 

((|) = 60 and 75°) 2-108 



Xtl 



FIGURE PAGE 

2-58 Diffraction Diagram for a Gap of Two Wave Lengths and a 

45° Approach Compared with that for a Gap Width /2 Wave 

Lengths with a 90° Approach 2-109 

2-59 Single Breakwater - Refraction - Diffraction Combined. . . 2-111 

2-60 Wave Reflection at Hamlin Beach, New York 2-112 

2-61 Standing Wave (Clapotis) System - Perfect Reflection from 

a Vertical Barrier - Linear Theory 2-114 

2-62 (H^/L^) max vs Beach Slope 2-119 

2-63 X2 vs Beach Slope for Various Values H^/L^ 2-119 

2-64 Wave of Limiting Steepness in Deep Water 2-121 

2-65 Breaker Height Index vs Deep Water Wave Steepness 2-122 

2-66 Dimensionless Depth at Breaking vs Breaker Steepness . . . 2-123 

2-67 Spilling Breaking Wave 2-125 

2-68 Plunging Breaking Wave 2-125 

2-69 Surging Breaking Wave 2-126 

2-70 Collapsing Breaking Wave 2-126 

3-1 Sample Wave Records 3-3 

3-2 Waves in a Coastal Region 3-4 

3-3 Theoretical and Observed Wave-Height Distributions .... 3-7 

3-4 Theoretical and Observed Wave-Height Distributions .... 3-8 

3-5 Theoretical Wave-Height Distributions 3-9 

3-6 Typical Wave Spectra from the Atlantic Coast 3-14 

3-7 Observed and Hindcasted Significant Wave Heights vs Time . 3-18 
3-8 Map of North Atlantic Grid Points, Ocean Weather Ship 

(OWS) Stations and Argus Island 3-19 

3-9 Surface Synoptic Chart for 0030Z, 27 October 1950 3-23 

3-10 Sample Plotted Report 3-25 

3-11 Geostrophic Wind Scale 3-26 

3-12 Possible Fetch Limitations 3-28 

3-13 Relation of Effective Fetch to Width-Length Ratio for 

Rectangular Fetches 3-31 

3-14 Computation of Effective Fetch for Irregular Shoreline . . 3-32 
3-15 Deepwater Wave Forecasting Curves (for Fetches of 1 to 

1000 miles) 3-36 

3-16 Deepwater Wave Forecasting Curves (for Fetches of 100 

to More than 1000 miles) 3-37 

3-17 Location of Wave Hindcasting Stations and Summary of 

Synoptic Meterological Observations (SSMO) Areas .... 3-41 

3-18 A Comparison of Shipboard Observations and Hindcasts . . . 3-43 

3-19 Decay Curves 3-44 

3-20 Travel Time of Swell Based on t^, = D/C^ 3-45 

3-21 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 5 Feet 3-47 

3-22 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 10 Feet 3-47 

3-23 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 15 Feet 3-48 



XIII 



FIGURE 



PAGE 



3-24 Forecasting Curves for Shallow- Water Waves; Constant 

Depth = 20 Feet 3-48 

3-25 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 25 Feet 3-49 

3-26 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 30 Feet 3-49 

3-27 Forecasting Curves for Shallow- Water Waves; Constant 

Depth = 35 Feet 3-50 

3-28 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 40 Feet 3-50 

3-29 Forecasting Curves for Shallow- Water Waves; Constant 

Depth = 45 Feet 3-51 

3-30 Forecasting Curves for Shallow-Water Waves; Constant 

Depth = 50 Feet 3-51 

3-31 Typical Hurricane Wave Spectra 3-53 

3-32 Composite Wave Charts 3-55 

3-33 Pressure and Wind Distribution in Model Hurricane 3-57 

3-34 Isolines of Relative Significant Wave Height for Slow 

Moving Hurricane 3-59 

3-35 Relationship for Friction Loss Over a Bottom of 

Constant Depth 3-64 

3-36 Typical Tide Curves Along Atlantic and Gulf Coasts .... 3-72 
3-37 Typical Tide Curves Along Pacific Coasts of the United 

States 3-73 

3-38 Sample Tsunami Records from Tide Gages 3-75 

3-39 Typical Water Level Variations in Lake Erie 3-77 

3-40 Long-Wave Surface Profiles 3-79 

3-41 Storm Surge and Observed Tide Chart 3-86 

3-42 High Water Mark Chart for Texas, Hurricane Carla, 

7-12 September 1961 3-88 

3-43 Notation and Reference Frame 3-93 

3-44 Storm Surge Chart 3-98 

3-45 Schematic of Forces and Responses for Bathystrophic 

Approximation 3-102 

3-46 Various Setup Components Over the Continental Shelf. . . . 3-107 

3-47 Track for Hurricane Camille, August 1969 3-110 

3-48 Foot Surface Isovels (knots), Hurricane Camille, 

August 1969 3-111 

3-49 Seabed Profile Used for Hurricane Camille, August 1969 . . 3-114 
3-50 Open Coast Surge Hydrograph, Hurricane Camille, 

August 1969 3-118 

3-51 Preliminary Estimate of Peak Surge 3-119 

3-52 Shoaling Factors on Gulf Coast 3-121 

3-53 Shoaling Factors on East Coast 3-122 

3-54 Correction Factor for Storm Motion 3-123 

3-55 Comparison of Observed and Computed Peak Surges (for 43 

Storms with a Landfall South of New England from 

1893 - 1957) 3-124 

3-56 Surge Profile Along Coast Hurricane Camille, 

August 1969 3-127 



XIV 



FIGURE PAGE 

3-57 Lake Surface Contours on Lake Okeechobee, Florida 

Hurricane, 26-27 August 1949 3-129 

3-58 Grid System 3-133 

3-59 Lake Erie 3-135 

3-60 Cross-Section Area and Surface Width - Lake Erie 3-136 

3-61 Mean Bottom Profile of Lake Erie 3-137 

3-62 Wind Speed and Direction for Lake Erie - Storm, March 1955. 3-158 
3-63 Wind Setup Hydrograph for Buffalo and Toledo - Storm, 

March 1955 3-141 

3-64 Wind Setup Profile for Lake Erie - Storm, March 1955. . . . 3-142 

4-1 Typical Profile Changes with Time, Westhampton Beach, N.Y.. 4-2 

4-2 Three Types of Shoreline 4-3 

4-3 Shoreline Erosion near Shipbottom, N.J 4-7 

4-4 Shoreline Accretion and Erosion Near Beach Haven, N.J.. . . 4-8 

4-5 Stable Shoreline Near Peahala, N.J 4-9 

4-6 Fluctuations in Location of Mean Sea Level Shoreline on 

Seven East Coast Beaches 4-12 

4-7 Grain Size Scales 4-14 

4-8 Example Size Distribution 4-17 

4-9 Sand Size Distribution Along the U.S. Atlantic Coast. . . . 4-23 
4-10 Mean Monthly Nearshore Wave Heights for Five Coastal 

Segments 4-31 

4-11 Mean Monthly Nearshore Wave Period for Five Coastal 

Segments 4-32 

4-12 Distribution of Significant Wave Heights from Coastal 

Wave Gages for 1-year Records 4-34 

4-13 Nearshore Current System Near LaJolla Canyon, California. . 4-44 

4-14 Typical Rip Currents, Ludlam Island, N.J 4-46 

4-15 Distribution of Longshore Current Velocities 4-47 

4-16 Measured versus Predicted Longshore Current Speed 4-49 

4-17 Coasts in the Vicinity of New York Bight 4-51 

4-18 Three Scales of Profiles, Westhampton, Long Island 4-55 

4-19 Unit Volume Change versus Time Between Surveys for Profiles 

on South Shore of Long Island 4-59 

4-20 Maximum Wave Induced Bottom Velocity as a Function of 

Relative Depth 4-62 

4-21 Maximum Bottom Velocity from Small Amplitude Theory .... 4-63 

4-22 Initiation of Ripple Motion 4-66 

4-23 Wave Conditions Producing Maximum Bottom Velocity of 

0.5 ft/sec 4-67 

4-24 Nearshore Bathymetry with Shore-Parallel Contours off 

Panama City, Florida 4-68 

4-25 Nearshore Bathymetry with Shore-Parallel Contours and 

Linear Bars off Manasquan, N.J 4-69 

4-26 Slow Accretion of Ridge-and-Runnel at Crane Beach, Mass.. . 4-76 

4-27 Rapid Accretion of Ridge-and-Runnel - Lake Michigan .... 4-77 
4-28 Typical Berm and Bar Profiles from Prototype Size 

Laboratory Wave Tank 4-79 



XV 



FIGURE PAGE 

4-29 Berra - Bar Criterion Based on Dimensionless Fall Time and 

Deep Water Steepness 4-82 

4-30 Berm - Bar Criterion Based on Dimensionless Fall Time and 

Height to Grain Size Ratio 4-83 

4-31 Fall Velocity of Quartz Spheres in Water as a Function of 

Diameter and Temperature 4-84 

4-32 Data Trends - Median Grain Size versus Foreshore Slope. . . 4-86 

4-33 Data - Median Grain Size versus Foreshore Slope 4-87 

4-34 Longshore Component of Wave Energy Flux in Dimensionless 

Form as a Function of Breaker Conditions 4-92 

4-35 Longshore Component of Wave Energy Flux as a Function of 

Deepwater Wave Conditions 4-94 

4-36 Transport Rate versus Energy Flux Factor for Field and 

Laboratory Conditions 4-99 

4-37 Relationship Between Wave Energy and Longshore Transport. . 4-100 
4-38 Longshore Transport Rate as a Function of Breaker Height 

and Breaker Angle 4-103 

4-39 Longshore Transport Rate as a Function of Deepwater Wave 

Height and Deepwater Angle 4-104 

4-40 Upper Limit on Longshore Transport Rates 4-109 

4-41 Typical Barrier Island Profile Shape 4-112 

4-42 Event Frequency per 100 Years the Stated Level is Equalled 

or Exceeded on the Open Coast, South Padre Island, Texas. 4-114 

4-43 Basic Example of Sediment Budget 4-120 

4-44 Erosion Within Littoral Zone During Uniform Retreat of an 

Idealized Profile 4-122 

4-45 Sediment Trapped Inside Old Drum Inlet, N.C 4-125 

4-46 Overwash on Portsmouth Island, N.C 4-126 

4-47 Growth of a Spit in to Deep Water, Sandy Hook, N.J 4-128 

4-48 Dunes Migrating Inland Near Laguna Point, California. . , . 4-130 

4-49 Materials Budget for Littoral Zone 4-132 

4-50 Summary of Example Problem Conditions and Results 4-135 

4-51 Variation of y with Distance Along Spit 4-136 

4-52 Growth of Sandy Hook, N.J., 1835-1932 4-141 

4-53 Transport Directions at New Buffalo Harbor Jetty on 

Lake Michigan 4-142 

4-54 Sand Accumulation at Point Mugu, California 4-143 

4-55 Tombolo and Pocket Beach at Greyhound Rock, California. . . 4-144 
4-56 A Nodal Zone of Divergence Illustrated by Sand Accumulation 

at Groins, South Shore Staten Island, N.Y 4-145 

4-57 South Shore of Long Island, N.Y., Showing Closed, 

Partially Closed and Open Inlets 4-146 

4-58 Four Types of Barrier Islands Offset 4-148 

4-59 Fire Island Inlet, New York - Overlapping Offset 4-149 

4-60 Old Drum Inlet, North Carolina - Negligible Offset 4-149 



XVI 



LIST OF TABLES 

TABLE PAGE 

1-1 Shoreline Characteristics 1-3 

2-1 Distribution of Wave Heights in a Short Train of Waves. . . 2-32 
2-2 Example Computations of Values of C,/C„ for Refraction 

Analysis 2-71 

3-1 Correction for Sea-Air Temperature 3-27 

3-2 Wind- Speed Adjustment, Nearshore 3-29 

3-3 Values of Kg or (H/H^) 3-63 

3-4 Computations for Wind Waves Over the Continental Shelf. . . 3-66 

3-5 Tidal Ranges 3-74 

3-6 Fluctuations in Water Levels - Great Lakes System 

(1860 through 1970) 3-76 

3-7 Short-Period Fluctuations in Lake Levels at Selected 

Gage Sites 3-78 

3-8 Highest and Lowest Water Levels 3-89 

3-9 Systems of Units for Storm Surge Computations 3-108 

3-10 Manual Surge Computations 3-116 

4-1 Seasonal Profile Changes on Southern California Beaches . . 4-10 

4-2 Density of Littoral Materials 4-19 

4-3 Minerals Occurring in Beach Sand 4-20 

4-4 Mean Wave Height at Coastal Localities of Conterminous 

United States 4-33 

4-5 Storm- Induced Beach Changes 4-72 

4-6 Longshore Transport Rates from U.S. Coasts 4-90 

4-7 Longshore Energy Flux ?i, for a Single Periodic Wave in 

Any Specified Depth 4-97 

4-8 Approximate Formulas for Computing Longshore Energy Flux 

Factor, Pj,g, Entering the Surf Zone 4-97 

4-9 Assumptions for P^g Formulas in Table 4-8 4-98 

4-10 Deepwater Wave Heights, in Percent by Direction, off East 

Facing Coast of Inland Sea 4-105 

4-11 Computed Longshore Transport for East - Facing Coast of 

Inland Sea 4-106 

4-12 Estimate of Gross Longshore Transport Rate for Shore of 

Inland Sea 4-110 

4-13 Classification of Elements in the Littoral Zone Sediment 

Budget 4-118 

4-14 Sand Budget of the Littoral Zone 4-133 



XVII 



CHAPTER 1 



INTRODUCTION 

TO 



COASTAL ENGINEERING 





^AA^At<c~ 



HALEIWA BEACH, OAHU, HAWAII - 23 August 1970 



CHAPTER 1 

INTRODUCTION TO COASTAL ENGINEERING 

1.1 INTRODUCTION TO THE SHORE PROTECTION MANUAL 

This Shore Protection Manual has been prepared to assemble in a 
single three-volume publication coastal-engineering practices for shore 
protection. "Coastal Engineering" is defined as the application of the 
physical and engineering sciences to the planning, design, and construc- 
tion of works to modify or control the interaction of the air, sea, and 
land in the coastal zone for the benefit of man and for the enhancement 
of natural shoreline resources. "Shore protection," as used in this 
Manual, applies to works designed to stabilize the shores of large bodies 
of water where wave action is the principal cause of erosion. Much of 
the material is applicable to the protection of navigation channels and 
harbors. 

The nature and degree of required shore-protection measures vary 
widely at different localities. Proper solution of any specific problem 
requires systematic and thorough study. The first requisite for such 
study is a clear definition of the problem and the objectives sought. 
The first factor to be determined is the cause of the problem. Ordinarily 
there will be more than one method of obtaining the immediate objective. 
Therefore, the long-term effects of each method should be studied. The 
immediate and long-term effects of each method should be evaluated not 
only within the problem area, but also in adjacent shore areas. All 
physical and environmental effects, advantageous and detrimental, should 
be considered in comparing annual costs and benefits to determine the 
justification of protection methods. 

Detailed summaries of applicable methods, techniques, and useful data 
pertinent to the solution of shore protection problems have been included 
in this Manual. 

By replacing Shore Proteatioriy Planning and Design with the Shore 
Protection Manual^ CERC is providing coastal engineers with an improved 
tool for solving shore-protection problems. The Manual is designed as 
an advanced text, but contains sufficient introductory material to allow 
a person with an engineering background to obtain an understanding of 
coastal phenomena and to solve related engineering problems. 

Chapter 1 presents a basic introduction to the subject. Chapter 2, 
"Mechanics of Wave Motion," treats wave theories, wave refraction and 
diffraction, wave reflection, and breaking waves. Chapter 3, "Wave and 
Water Level Predictions," discusses wave forecasting, hurricane waves, 
storm surge, and water level fluctuations. Chapter 4, "Littoral Proc- 
esses, y treats the characteristics and sources of littoral material 
nearshore currents, littoral transport, and sand budget techniques. 
Chapter 5, "Planning Analyses," treats the functional planning of shore- 
protection measures. Chapter 6, "Structural Features," illustrates the 



l-l 



functional design of various structures. Chapter 7, "Structural 
Design — Physical Factors," treats tke effects of environmental forces 
on the design of protective works. Chapter 8, "Engineering .Analysis — 
Case Study," presents a series of calculations for the preliminary 
design of an offshore island facility in the mouth of the Delaware 
Bay. 

Each chapter contains its own bibliography. This Manual concludes 
with four appendixes. Because the meanings of coastal engineering terms 
differ from place to place, the reader is urged to use Appendix A, 
Glossary of Terms, that defines the terms used in this Manual. Appen- 
dix B lists the symbols used. Appendix C is a collection of miscella- 
neous tables and plates that supplement the material in the chapters. 
Appendix D is the subject index. 

1,2 THE SHORE ZONE 

Table 1-1 summarizes regional shoreline characteristics. The infor- 
mation obtained from the "Report on the National Shoreline Study," by the 
Department of the Army, Corps of Engineers (1971), indicates that of the 
total 84,240 miles of U.S. shoreline, there are 34,520 miles (41 percent) 
of exposed shoreline and 49,720 miles (59 percent) of sheltered shoreline 
(i.e., in bays, estuaries and lagoons). About 20,500 miles of the shore- 
line (or 24 percent of the total) are eroding. Of the total length of 
shoreline, exclusive of Alaska (36,940 miles), about 12,150 miles (33 
percent) have beaches; the remaining 24,790 miles have no beach. 

1.21 NATURAL BEACH PROTECTION 

Where the land meets the ocean at a sandy beach, the shore has 
natural defenses against attack by waves, currents and storms. First 
of these defenses is the sloping nearshore bottom that causes waves to 
break offshore, dissipating their energy over the surf zone. The pro- 
cess of breaking often creates an offshore bar in front of the beach 
that helps to trip following waves. The broken waves re-form to break 
again, and may do this several times before finally rushing up the beach 
foreshore. At the top of wave uprush a ridge of sand is formed. Beyond 
this ridge, or crest of the berm, lies the flat beach berm that is 
reached only by higher storm waves. A beach profile and its related 
terminology are shown in Figure 1-1. 

1.22 NATURAL PROTECTIVE DUNES 

Winds blowing inland over the foreshore and berm move sand behind the 
beach to form dunes. (See Figures 1-2 and 1-3.) Grass, and sometimes 
bushes and trees, grow on the dunes, and the dunes become a natural levee 
against sea attack. Dunes are the final natural protection line against 
wave attack, and are also a reservoir for storage of sand against storm 
waves . 



1-2 







Table 


1-1. Shore 


ine Characteristics 








Shoreline 


Shore Change 


Shoreline 


Region 


Total 
(miles) 


Exposed 
(miles) 


Sheltered 
(nules) 


Eroding 
(miles) 


Non-Eroding 
(miles) 


With 
Beach 

(miles) 


Without 
Beach 
(miles) 


North 
Atlantic 


8,620 


4,730 


3,890 


7,460 


1,160 


2,320 


6,300 


South 
Atlantic- 
Gulf 


14,620 


2,470 


12,150, 


2,820 


11.800 


3,600 


11.020 


Lower 
Mississippi 


1,940 


810 


1,130 


1,580 


360 


830 


1.110 


Texas Gulf 


2,500 


370 


2.130 


360 


2,140 


380 


2.120 


Great Lakes 


3,680 


3,020 


660 


1,260 


2,420 


2,110 


1.570 


California 


1,810 


1,320 


490 


1,550 


260 


680 


1,130 


North 
Pacific 


2,840 


650 


2,190 


260 


2.580 


2,050 


790 


Hawaii 


930 


900 


30 


110 


820 


180 


750 


Total 


36,940 


14,270 


22.670 


15.400 


21.540 


12.150 


24.790 


Alaska 


47,300 


20.250 


27.050 


5.100 


42.200 


Unknown 


Unknown 


Total 
National 


84.240 


34.520 


49,720 


20.500 




63,740 


12.150 


24,790 



From: Report on National Shoreline Study, Department of the Army, Corps of Engineers, August 1971. 




Figure 1-1. Beach Profile - Related Terms 



-3 




Figure 1-2. Sand Dunes along the South Shore of Lake Michigan 





I^B 


%yv 


^ff?!??;'?^^ 


/ t^P'C^^^BBip-^?'^*'' it^FSs?^S?3r«435^ **''' 


■, ^ 




■ J.. 






Figure 1-3. Sand Dune, Honeyman State Park, Oregon 



1-4 



1.23 BARRIER BEACHES, LAGOONS AND INLETS 

In some areas, an additional natural protection for the mainland is 
provided in the form of barrier beaches. (See Figure 1-4.) Nearly all 
of the U.S. east coast from Long Island to Mexico is comprised of bar- 
rier beaches. These are long narrow islands or spits lying parallel to 
the shoreline. Barrier beaches generally enclose shallow lagoons that 
separate the mainland from the ocean. During severe storms these barrier 
beaches absorb the brunt of the wave attack. When barrier-beach dunes 
are breached, the result may be the cutting of an inlet. The inlet per- 
mits sand to enter the lagoon and settle to the bottom, removing sand 
from the beach. 

1.24 STORM ATTACK 

Diiring storms, strong winds generate high waves. Storm surge and 
waves may raise the water level near the shore. If storm surge does 
occur, large waves can then pass over the offshore bar formation without 
breaking. If the storm occurs at high tide, storm surge super-elevates 
the water, and some waves may break on the beach or even at the base of 
the dunes. After a storm or storm season, natural defenses may again be 
re-formed by normal wave and wind action. 

lo25 ORIGIN AND MOVEMENT OF BEACH SANDS 

Most of the sands of the beaches and nearshore slopes are normally 
small, resistant rock particles that have traveled many miles from in- 
land mountains. When the sand reaches the shore, it is moved alongshore 
by waves and littoral currents. This alongshore transport is a constant 
process, and great volumes may be transported. In most coastal segments 
the direction of movement changes as direction of wave attack changes. 

1.3 THE SEA IN MOTION 

1.31 TIDES AND WINDS 

The motions of the sea originate in the gravitational effects of the 
sun, the moon, and earth; and from air movements or winds caused by dif- 
ferential heating of the earth. 

The moon, and to a lesser extent the sun, creates ocean tides by 
gravitational forces. These forces of attraction, and the fact that the 
sun, moon, and earth are always in motion with relation to each other, 
cause waters of ocean basins to be set in motion. These tidal motions of 
water masses are a form of very long period wave motion, resulting in a 
rise and fall of the water surface at a point. There are normally two 
tides per day, but some localities have only one per day. 

1.32 WAVES 

The familiar waves of the ocean 'are wind waves generated by winds 
blowing over water. They may vary in size from ripples on a pond to 



1-5 




I -6 



large ocean waves as high as 100 feet. (See Figure 1-5.) Wind waves 
cause most of the damage to the ocean coasts. Another type of wave, 
the tsunami, is created by earthquakes or other tectonic disturbances 
on the ocean bottom. Tsunamis have caused spectacular damage at times, 
but fortunately, major tsunamis do not occur frequently. 

Wind waves are of the type known as oscillatory waves, and are 
usually defined by their height, length, and period. (See Figure 1-6.) 
Wave height is the vertical distance from the top of the crest to the 
bottom of the trough. Wavelength is the horizontal distance between 
successive crests„ Wave period is the time between successive crests 
passing a given point. 

As waves propagate in water, only the form and part of the energy 
of the waves move forward; the water particles remain. 

The height, length, and period of wind waves are determined by the 
fetch (the distance the wind blows over the sea in generating the waves), 
the wind speed, the length of time the wind blows, and the deoay distance 
(the distance the wave travels after leaving the generating area). Gener- 
ally, the longer the fetch, the stronger the wind; and the longer the 
time the wind blows, the larger the waves. The water depth, if shallow 
enough, will also affect the size of wave generated. The wind simul- 
taneously generates waves of many heights, lengths, and periods as it 
blows over the sea. 

If winds of a local storm blow toward the shore, the generated 
waves will reach the beach in nearly the form in which they are gener- 
ated. Under these conditions, the waves are steep; that is, the wave- 
length is 10 to 20 times the wave height. Such waves are called seas. 
If waves are generated by a distant storm, they may travel through 
hundreds or even thousands of miles of calm areas before reaching the 
shore. Under these conditions, waves decay - short, steep waves are 
eliminated, and only relatively long, low waves reach the shore. Such 
waves have lengths from 30 to more than 500 times the wave height, and 
are called swell. 

1.33 CURRENTS AND SURGES 

Currents are created in oceans and adjacent bays and lagoons when 
water in one area becomes higher than water in another area. Water in 
the higher area flows toward the lower area, creating a current. Some 
causes of differences in the elevation of the water surface in the oceans 
are tides, wind, waves breaking on a beach, and streams. Changes in water 
temperature or salinity cause changes in water density that may also pro- 
duce currents. 

Wind creates currents because, as it blows over the water surface, 
it creates a stress on surface water particles, and starts these parti- 
cles moving in the direction in which the wind is blowing. Thus, a sur- 
face current is created. When such a current reaches a barrier, such as 
coastline, water tends to pile up against the land. In this way, wind 



1-7 




li^Mirc l-fi. Large Wuve:; lii'caking ovci- u lireakwater 



Oifi'clion of Wave Irovel 



Wove Crest -^ 





Crest Lcngtti 
Region 



TrouQti Lcnqtii 
Region 



■Stillwoler level 



d^ Oeptti 



Oceon Bottom- 






Figure 1-0. W.ivc Cliar.Kt c-ri sties 



1-8 



setup or storm surges are created by strong winds. The height of stomi 
surge depends on wind velocity and dii'ection, fetch, water depth, ajid 
ncarshore slope. In violent storms, storm surge may raise sea level at 
the shore as much as 20 feet. In the United States, larger surges occur 
on the Gulf coast because of the shallower and broader shelf off that 
coast compared to the shelves off the Atlantic and Pacific coasts. Storm 
surges may also be increased by a funneling effect in converging estuaries. 

When waves approach the beach at an angle, they create a current in 
shallow water parallel to the shore, known as the lotuji^hovc c-'uvveyit. This 
current, under certain conditions, may turn and run out to sea in what is 
known as a rip aurvent. 

1.34 TIDAL CURRENTS 

If water level rises :uid falls at an area, then water must flow into 
and out of the area. Significant currents generated by tides occur at in- 
lets to lagoons and bays or at entrances to harbors. At such constricted 
places, tidal currents generally flow in when the tide is rising (flood 
tide) and flow out as the tide falls (ebb tide). Exceptions can occur 
at times of high river discharge or strong winds, or when density cur- 
rents are an important part of the current system. 

In addition to creating currents, tides const juitly chaiige the level 
at which waves attack the bcnclu 

1.4 THE BEHAVIOR OF BEACHES 

1.41 BEACH COMPOSITION 

The size and character of sediments on a beacli ai"e related to forces 
to which the beach is exposed and the type of material available at the 
shore. Most beaches are composed of fine or coarse sand and. in some 
areas, of small stones called shingle or gravel. This material is sup- 
plied to the beach zone by streams, by erosion of the shores caused by 
waves and currents and, in some cases, by onshore movement of material 
from deeper water. Clay and silt do not usually remain on ocean beaches 
because the waves create such turbulence in the water along the shore 
that these fine materials ;ire kept in suspension. It is only after nnw- 
ing away from the beaches into quieter or deeper water that these fine 
particles settle out and deposit on the bottom. 

1.42 BEACH CHARACTERISTICS 

Characteristics of a beach are usually described in terms of average 
size of the sand particles that make up the beach, range and distribution 
of sizes of those particles, sand composition, elevation and width of berm, 
slope or steepness of the foreshore, the existence (or lack) of a bar, and 
the general slope of the inshore zone fronting the beach. Generally, the 
larger the sand particles tlic steeper the beach slope. Beaches witli 
gently sloping foreshores and inshore zones usually have a prepondor.ince 



1-9 



of the finer sizes of sand. Daytona Beach, Florida, is a good example 
of a gently sloping beach composed of fine sand. 

lo43 BREAKERS 

As a wave moves toward shore, it reaches a depth of water so shallow 
that the wave collapses or breaks. This depth is equal to about 1.3 times 
the wave height. Thus a wave 3 feet high will break in a depth of about 
4 feet. Breaking can occur in several different ways (plunging, spilling, 
surging, or collapsing). Breaking results in a dissipation of the energy 
of the wave and is manifested by turbulence in the water. This turbulence 
stirs up the bottom materials. For most waves, the water travels forward 
after breaking as a foaming, turbulent mass, expending most of its remain- 
ing energy in a rush up the beach slope. 

1.44 EFFECTS OF WIND WAVES 

Wind waves affect beaches in two major ways. Short steep waves, 
which usually occur during a storm near the coast, tend to tear the 
beach down. (See Figure 1-7.) Long swells, which originate from dis- 
tant storms, tend to rebuild the beaches. On most beaches, there is a 
constant change caused by the tearing away of the beach by local storms 
followed by gradual rebuilding by swells. A series of violent local 
storms in a short time can result in severe erosion of the shore if 
there is not enough time between storms for swells to rebuild the 
beaches. Alternate erosion and accretion of beaches may be seasonal 
on some beaches; the winter storms tear the beach away, and the summer 
swells rebuild it. Beaches may also follow long-term cyclic patterns. 
They may erode for several years, and then accrete for several years. 

1.45 LITTORAL TRANSPORT 

Littovat transport is defined as the movement of sediments in the 
nearshore zone by waves and currents and is divided into two general 
classes; transport parallel to the shore (longshore transport) and 
transport perpendicular to the shore (on shore- offshore transport). This 
transport is distinguished from the material moved, which is called 
littoral drift. 

Onshore -offshore transport is determined primarily by wave steepness, 
sediment size, and beach slope. In general, high steep waves move material 
offshore, and low waves of long period (low steepness waves) move material 
onshore. This onshore-offshore process associated with storm waves is 
illustrated in Figure 1-7. 

Longshore transport results from the stirring up of sediment by the 
breaking wave, and the movement of this sediment by the component of the 
wave in an alongshore direction, and by the longshore current generated 
by the breaking wave. The direction of longshore transport is directly 
related to the direction of wave approach, and the angle of the wave to 
the shore. Thus, due to the variability of wave approach, longshore 



l-IO 



•Dune Crest 




Crest ■■:■■.■:■■.■ I I 

Lowering Profile C - Storm wove ottock •••••••• ■•"■••:■:•'\•■:'K•■V•:^;?>:3J^^^ MLW 

of foredune i^o^ 

Crest 



ACCRETION 




Profile A 




Profile D - After storm wove ottock, 
normol wove oction 



M.L.W. 



ACCRETION 




Profile A 



Figure 1-7, Schematic Diagram of Storm Wave Attack on Beach and Dune 



l-ll 



transport direction can vary from season to season, day to day or hour 
to hour. These reversals of transport direction are quite common for 
most United States shores. Direction may vary at random, but in most 
areas the net effect is seasonal. 

The rate of longshore transport is dependent on both angle of wave 
approach, and wave energy. Thus, high storm waves will generally move 
more material per unit time than low waves. However, if low waves 
exist for a much longer time than do high waves, the low waves may be 
more significant in moving sand than the high waves. 

Because reversals in transport direction occur, and because different 
types of waves transport material at different rates, two components of 
the longshore transport rate become important. The first is the net rate, 
the net amount of material passing a particular point in the predominant 
direction during an average year. The second component is the gross rate, 
the total of all material moving past a given point in a year regardless 
of direction. Most shores consistently have a net annual longshore trans- 
port in one direction. Determining the direction and average net and 
gross annual amount of longshore transport is important in developing 
shore protection plans. 

In landlocked water of limited extent, such as the Great Lakes, a 
longshore transport rate in one direction can normally be expected to be 
no more than about 150,000 cubic yards per year. For open ocean coasts, 
the net rate of transport may vary from 100,000 to more than 2 million 
cubic yards per year. The rate depends on the local shore conditions 
and shore alignment as well as the energy and direction of wave action. 

1.46 EFFECT OF INLETS ON BARRIER BEACHES 

Inlets may have significant effects on adjacent shores by interrupt- 
ing the longshore transport and trapping onshore- offshore moving sand. 
On ebb current, sand moved to the inlet by waves is carried a short dis- 
tance out to sea and deposited on an outer bar. When this bar becomes 
large enough, the waves begin to break on it, and sand again begins to 
move over the bar back toward the beach. On the flood tide, when water 
flows through the inlet into the lagoon, sand in the inlet is carried a 
short distance into the lagoon and deposited. This process creates shoals 
in the landward end of the inlet known as middlegroimd shoals or inner 
bars. Later, ebb flows may bring some of the material in these shoals 
back to the ocean, but some is always lost from the stream of littoral 
drift and thus from the downdrift beaches. In this way, tidal inlets 
may store sand and reduce the supply of sand to adjacent shorelines. 

1.47 IMPACT OF STOM/IS 

Hurricanes or severe storms moving over the ocean near the shore may 
greatly change beaches. Strong winds of a storm often create a storm surge. 
This surge raises the water level and exposes to wave attack higher parts 
of the beach not ordinarily vulnerable to waves. Such storms also generate 



1-12 



large, steep waves. These waves carry large quantities of sand from the 
beach to the nearshore bottom. Land structures, inadequately protected 
and located too close to the water, are then subjected to the forces of 
waves and may be damaged or destroyed. Low- lying areas next to the ocean, 
lagoons, and bays are often flooded by storm surge. Storm surges are 
especially damaging if they occur concurrently with astronomical high tide. 

Beach berms are built naturally by waves to about the highest eleva- 
tion reached by normal storm waves. Berms tend to absorb the wave energy; 
however, overtopping permits waves to reach the dunes or bluffs in back of 
the beach and damage unprotected upland features. 

When storm waves erode the berm and carry the sand offshore, the pro- 
tective value of the berm is reduced and large waves can overtop the beach. 
The width of the berm at the time of a storm is thus an important factor 
in the amount of upland damage a storm can inflict. 

Notwithstanding changes in the beach that result from storm-wave 
attack, a gently sloping beach of adequate width and height is the most 
effective method known for dissipating wave energy. 

1.48 BEACH STABILITY 

Although a beach may be temporarily eroded by storm waves and later 
partly or wholly restored by swells, and erosion and accretion patterns 
may occur seasonally, the long-range condition of the beach - whether 
eroding, stable or accreting - depends on the rates of supply and loss 
of littoral material. The shore accretes or progrades when the rate of 
supply exceeds the rate of loss. The shore is considered stable (even 
though subject to storm and seasonal changes) when the long-term rates 
of supply and loss are equal. 

1.5 EFFECTS OF MAN ON THE SHORE 

1.51 ENCROACHMENT ON THE SEA 

During the early days of the United States, natural beach processes 
continued to mold the shore as in ages past. As the country developed, 
activity in the shore area was confined principally to harbor areas. 
Between harbor areas, development along the shore progressed slowly as 
small, isolated, fishing villages. As the national economy grew, im- 
provements in transportation brought more people to the beaches. Gradu- 
ally, extensive housing, commercial, recreational and resort developments 
replaced fishing villages as the predominant coastal manmade features. 
Examples of this development are Atlantic City and Miami Beach. 

Numerous factors control the growth of development at beach areas, 
but undoubtedly the beach environment is the development's basic asset. 
The desire of visitors, residents, and industries to find accommodations 
as close to the ocean as possible has resulted in man's encroachment on 
the sea. 



1-13 



There are places where the beach has been gradually widened, as well 
as narrowed, by natural processes over the years. This is evidenced by 
lighthouses and other structures that once stood on the beach, but now 
stand hundreds of feet inland. 

In their eagerness to be as close as possible to the water, developers 
and property owners often forget that land comes and goes, and that land 
which nature provides at one time may later be reclaimed by the sea. Yet 
once the seaward limit of a development is established, this line must be 
held if large investments are to be preserved. This type of encroachment 
has resulted in great monetary losses due to storm damage, and in ever- 
increasing costs of shore protection. 

lo52 NATURAL PROTECTION 

While the sloping beach and beach berm are the outer line of defense 
to absorb most of the wave energy, dunes are the last zone of defense in 
absorbing the energy of storm waves that overtop the berm. Although dunes 
erode during severe storms, they are often substantial enough to afford 
complete protection to the land behind them. Even when breached by waves 
of a severe storm, dunes may gradually rebuild naturally to provide pro- 
tection during future storms. Continuing encroachment on the sea with 
manmade development has often taken place without proper regard for the 
protection provided by dunes. Large dune areas have been leveled to make 
way for real estate developments, or have been lowered to permit easy 
access to the beach. Where there is inadequate dune or similar protection 
against storm waves, the storm waters may wash over low- lying land, mov- 
ing or destroying everything in their path, as illustrated by Figure 1-8. 

1.53 SHORE PROTECTION METHODS 

Where beaches and dunes protect shore developments, additional pro- 
tective works may not be required. However, when natural forces do 
create erosion, storm waves may overtop the beach and damage backshore 
structures. Manmade structures -must then provide protection. In gene- 
ral, measures designed to stabilize the shore fall into two classes: 
structures to prevent waves from reaching erodible material (seawalls, 
bulkheads, revetments); and an artificial supply of beach sand to make 
up for a deficiency in sand supply through natural processes. Other 
manmade structures, such as groins and jetties, are used to retard the 
longshore transport of littoral drift. These may be used in conjunction 
with seawalls or beachfills or both. 

Separate protection for short reaches of eroding shores (as an indi- 
vidual lot frontage) within a larger zone of eroding shore, is difficult 
and costly. Such protection often fails at its flanks as the adjacent 
unprotected shores continue to recede. Partial or inadequate protective 
measures may even accelerate erosion of adjacent shores. Coordinated 
action under a comprehensive plan that considers erosion processes over 
the full length of the regional shore compartment is much more effective 
and economical. 



1-14 





2 



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I— 1 
(A 



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E 
nj 
Q 

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I -15 



1.54 BULKHEADS, SEAWALLS AND REVETMENTS 

Protection on the upper part of the beach which fronts backshore 
development is required as a partial substitute for the natural pro- 
tection that is lost when the dunes are destroyed. Shorefront owners 
have resorted to shore armoring by wave- resist ant walls of various 
types. A vertical wall in this location is known as a bulkhead, and 
serves as a secondary line of defense in major storms. Bulkheads are 
constructed of steel, timber, or concrete piling. For ocean-exposed 
locations, bulkheads do not provide a long-term solution, because a 
more substantial wall is required as the beach continues to recede and 
larger waves reach the structure. Unless combined with other types of 
protection, the bulkhead must be enlarged into a massive seawall capable 
of withstanding the direct onslaught of the waves. Seawalls may have 
vertical, curved or stepped faces. While seawalls may protect the up- 
land, they can create a local problem. Downward forces of water created 
by waves striking the wall, can rapidly remove sand from in front of the 
wall. A stone apron is often necessary to prevent excessive scouring 
and undermining. 

A revetment armors the slope face of a dune or bluff. It is usually 

composed of one or more layers of stone or is of concrete construction. 

This sloping protection dissipates wave energy with less damaging effect 

on the beach than waves striking vertical walls. 

1.55 BREAKWATERS 

Beaches and bluffs or dunes can be protected by an offshore break- 
water that reduces the wave energy reaching the shore. However, offshore 
breakwaters are usually more costly than onshore structures, and are sel- 
dom built solely for shore protection. Offshore breakwaters are construc- 
ted mainly for navigation purposes. A breakwater protecting a harbor 
area provides shelter for boats. Breakwaters have both beneficial and 
detrimental effects on the shore. All breakwaters reduce or eliminate 
wave action and thus protect the shore immediately behind them. Whether 
offshore or shore-connected, the elimination of wave action reduces long- 
shore transport, obstructing the movement of sand along the shore and 
starving the downdrift beaches. 

At a harbor breakwater, the sand stream generally can be restored by 
pumping sand through a pipeline from the side where sand accumulates to 
the eroded downdrift side. This type of operation has been in use for 
many years at Santa Barbara, California. 

Even without a shore arm, an offshore breakwater reduces wave action 
and creates quiet water between it and the shore. In the absence of wave 
action to move sand, it is deposited and builds the shore seaward toward 
the breakwater. The buildup serves as a barrier which also blocks the 
movement of littoral materials. If the offshore breakwater is placed 
immediately updrift from a navigation opening, the structure impounds 
sand, prevents it from entering the navigation channel, and affords 



1-16 



shelter for a floating dredge to pump the impounded material across the 
navigation opening back onto the downdrift beach. This method is used 
at Channel Island Harbor near Port Hueneme, California. 

1.56 GROINS 

The groin is a barrier-type structure that extends from the backshore 
into the littoral zone. The basic purposes of a groin are to interrupt 
longshore sand movement, to accumulate sand on the shore, or to retard 
sand losses. Trapping of sand by a groin is done at the expense of the 
adjacent downdrift shore unless the groin or groin system is artificially 
filled with sand to its entrapment capacity from other sources. To reduce 
the potential for damage to property downdrift of a groin, some limitation 
must be imposed on the amount of sand permitted to be impounded on the 
updrift side. Since more and more shores are being protected, and less 
and less sand is available as natural supply, it is now desirable, and 
frequently necessary, to place sand artificially to fill the area between 
the groins, thereby ensuring a more or less uninterrupted passage of the 
sand to the downdrift shores. 

Groins have been constructed in various configurations using timber, 
steel, concrete or rock. Groins can be classified as high or low, long 
or short, permeable or impermeable, and fixed or adjustable. 

A high groin, extending through the breaking zone for ordinary or 
moderate storm waves, initially entraps nearly all of the longshore 
moving sand within that intercepted area until the areal pattern or sur- 
face profile of the accumulated sand mass allows sand to pass around the 
seaward end of the groin to downdrift shores. Low groins (top profile no 
higher than that of desired beach dimensions) function like high groins, 
except that sand also passes over the top of the structure. Permeable 
groins permit some of the wave energy and moving sand to pass through the 
structure. 

1.57 JETTIES 

Jetties are generally employed at inlets in connection with naviga- 
tion improvements. When sand being transported along the coast by waves 
and currents arrives at an inlet, it flows inward on the flood tide to 
form an inner bar, and outward on the ebb tide to form an outer bar. 
Both formations are harmful to navigation through the inlet, and must be 
controlled to maintain an adequate navigation channel. The jetty is sim- 
ilar to the groin in that it traps sand moving along the beach. Jetties 
are usually constructed of steel, concrete, or rock. The jetty type 
depends on foundation conditions, wave climate, and economic considera- 
tions. Jetties are much larger than groins, since jetties sometimes ex- 
tend from the shoreline seaward to a depth equivalent to the channel 
depth desired for navigation purposes. To be efficient in maintaining 
the channel, the jetty must be high enough to completely obstruct sand 
movement . 

1-17 



Jetties aid navigation by reducing movement of sand into the channel, 
by stabilizing the location of the channel, and by shielding vessels from 
waves. Sand is impounded at the updrift jetty, and the supply of sand to 
the shore downdrift from the inlet is reduced, thus causing erosion of 
that shore. Before the installation of a jetty, nature supplied sand by 
transporting it across the inlet intermittently along the outer bar to the 
downdrift shore. 

To eliminate undesirable downdrift erosion, some projects provide for 
dredging the sand impounded by the updrift jetty and pumping it through a 
pipeline (bypassing the inlet) to the eroding beach. This provides an 
intermittent flow of sand to nourish the downdrift beach, and also prevents 
shoaling of the entrance channel. 

A more recent development for sand bypassing provides a low section 
or weir in the updrift jetty over which sand moves into a sheltered pre- 
dredged, deposition basin. By dredging the basin periodically, deposition 
in the channel is reduced or eliminated. The dredged material is normally 
pumped across the inlet to provide nourishment for the downdrift shore. 
A weir jetty of this type at Masonboro Inlet, North Carolina, is shown in 
Figure 1-9. 

1.58 BEACH RESTORATION AND NOURISHMENT 

As previously stated, beaches are very effective in dissipating wave 
energy. When maintained to adequate dimensions, they afford protection 
for the adjoining backshore. Therefore, a protective beach is classed as 
a shore-protection structure. When studying an erosion problem, it is gen- 
erally advisable to investigate the feasibility of mechanically or hydrau- 
lically placing borrow material on the shore to form and maintain an ade- 
quate protective beach. The method of placing beach fill to ensure sand 
supply at the required replenishment rate is important. Where stabiliza- 
tion of an eroding beach is the problem, suitable beach material may be 
stockpiled at the updrift sector of the problem area. The establishment 
and periodic replenishment of such a stockpile is termed artificial 
beaoh nourishment. To restore an eroded beach and stabilize it at the 
restored position, fill is placed directly along the eroded sector, and 
then the beach is artificially nourished by the stockpiling method. 

When conditions are suitable for artificial nourishment, long reaches 
of shore may be protected by this method at a relatively low cost per 
linear foot of protected shore. An equally important advantage is that 
artificial nourishment directly remedies the basic cause of most erosion 
problems - a deficiency in natural sand supply - and benefits rather than 
damages the adjacent shore. An added consideration is that the widened 
beach has value as a recreation feature. A project for beach restoration 
with an artificial dune for protection against hurricane wave action, com- 
pleted in 1965 at Wrightsville Beach, North Carolina, is shown in Figure 

l-lOo 

1-18 




Figure 1-9. Weir Jetty at Masonboro Inlet, North Carolina 



1-19 




I -20 



Sometimes structures must be provided to protect dunes, to maintain 
a specific beach dimension, or to reduce nourishment requirements. In 
each case, the cost of such structures must be weighed against the bene- 
fits they would provide. Thus, measures to provide and keep a wider pro- 
tective and recreational beach for a short section of an eroding shore 
would require excessive nourishment without supplemental structures such 
as groins to reduce the rate of loss of material from the widened beach. 
A long, high terminal groin or jetty is frequently justified at the down- 
drift end of a beach restoration project to reduce losses of fill into an 
inlet and to stabilize the lip of the inlet. 

1.6 CONSERVATION OF SAND 

Experience and study have demonstrated that sand from dunes, beaches, 
and nearshore areas is the best material available naturally in suitable 
form to protect shores. Where sand is available in abundant quantities, 
protective measures are greatly simplified and reduced in cost. When 
dunes and broad, gently sloping beaches can no longer be provided, it is 
necessary to resort to alternative structures, and the recreational attrac- 
tion of the seashore is lost or greatly diminished. 

Sand is a diminishing natural resource. Sand was once available to 
our shores in adequate supply from streams, rivers and glaciers, and by 
coastal erosion. Now cultural development in the watershed areas and along 
previously eroding shores has progressed to a stage where large areas of 
our coast now receive little or no sand through natural geological pro- 
cesses. Continued cultural development in both inland and shore areas 
tends to further reduce coastal erosion with resulting reduction in sand 
supply to the shore. It thus becomes apparent that sand must be conserved. 
This does not mean local hoarding of beach sand at the expense of adjoin- 
ing areas, but rather the elimination of wasteful practices and the pre- 
vention of losses from the shore zone whenever feasible. 

Fortunately, nature has provided extensive stores of beach sand in 
bays, lagoons, estuaries and offshore areas that can be used as a source 
of beach and dune replenishment where the ecological balance will not be 
disrupted. Massive dune deposits are also available at some locations, 
though these must be used with caution to avoid exposing the area to flood 
hazard. These sources are not always located in the proper places for 
economic utilization, nor will they last forever. When they are gone, we 
must face increasing costs for the preservation of our shores. Offshore 
sand deposits will probably become the most important source in the future. 

Mechanical bypassing of sand at coastal inlets is one means of conser- 
vation that will come into increasing practice. Mining of beach sand for 
commercial purposes, formerly a common procedure, is rapidly being reduced 
as coastal communities learn the need for regulating this practice. Modem 
hopper dredges, used for channel maintenance in coastal inlets, are being 
equipped with a pump-out capability so their loads can be discharged near 
the shore instead of being dumped at sea. On the California coast, where 
large volumes of sand are lost into deep submarine canyons near the shore. 



1-21 



facilities are being considered that will trap the sand before it reaches 
the canyon and transport it mechanically to a point where it can resume 
normal longshore transport. Dune planting with appropriate grasses and 
shmbs reduces landward windbome losses and aids in dune preservation. 

Sand conservation is an important factor in the preservation of our 
seacoasts, and must be included in long-range planning. Protection of 
our seacoasts is not a simple problem; neither is it insurmountable. It 
is a task and a responsibility that has increased tremendously in impor- 
tance in the past 50 years, and is destined to become a necessity in future 
years. While the cost will mount as time passes, it will be possible 
through careful planning, adequate management, and sound engineering to 
do the job properly and economically. 



1-22 



CHAPTER 2 



MECHANICS 



OF 

WAVE MOTION 




LEO CARRILLO STATE BEACH, CALIFORNIA - July 1968 



CHAPTER 2 
MECHANICS OF WAVE MOTION 

2.1 INTRODUCTION 

The effects of water waves are of paramount importance in the field 
of coastal engineering. Waves are the major factor in determining the 
geometry and composition of beaches, and significantly influence planning 
and design of harbors, waterways, shore protection measures, coastal 
structures, and other coastal works. Surface waves generally derive their 
energy from the winds. A significant amount of this wave energy is finally 
dissipated in the nearshore region and on the beaches. 

Waves provide an important energy source for forming beaches; assort- 
ing bottom sediments on the shoreface; transporting bottom materials on- 
shore, offshore, and alongshore; and for causing many of the forces to 
which coastal structures are subjected. An adequate understanding of the 
fundamental physical processes in surface wave generation and propagation 
must precede any attempt to understand complex water motion in the near- 
shore areas of large bodies of water. Consequently, an understanding of 
the mechanics of wave motion is essential in the planning and design of 
coastal works. 

This chapter presents an introduction to surface wave theories. 
Surface and water particle motion, wave energy, and theories used in 
describing wave transformation due to interaction with the bottom and 
with structures are described. The purpose is to provide an elementary 
physical and mathematical understanding of wave motion, and to indicate 
limitations of selected theories. A number of wave theories have been 
omitted. References are cited to provide information on theories not 
discussed and to supplement the theories presented. 

The reader is cautioned that man's ability to describe wave phenomena 
is limited, especially when the region under consideration is the coastal 
zone. Thus, the results obtained from the wave theories presented should 
be carefully interpreted for application to actual design of coastal struc- 
tures or description of the coastal environment. 

2.2 WAVE MECHANICS 

2.21 GENERAL 

Waves in the ocean often appear as a confused and constantly changing 
sea of crests and troughs on the water surface because of the irregularity 
of wave shape and the variability in the direction of propagation. This 
is particularly true while the waves are under the influence of the wind. 
The direction of wave propagation can be assessed as an average of the 
directions of individual waves. A description of the sea surface is 
difficult because of the interaction between individual waves. Faster 
waves overtake and pass through slower ones from various directions. 
Waves sometimes reinforce or cancel each other by this interaction, and 
often collide with each other and are transformed into turbulence, and 

2-1 



spray. When waves move out of the area where they are directly affected 
by the wind, they assume a more ordered state with the appearance of 
definite crests and troughs and with a more rhythmic rise and fall. These 
waves may travel hundreds or thousands of miles after leaving the area in 
which they were generated. Wave energy is dissipated internally within 
the fluid by interaction with the air above, by turbulence on breaking, 
and at the bottom in shallow depths. 

Waves which reach coastal regions expend a large part of their energy 
in the nearshore region. As the wave nears the shore, wave energy may be 
dissipated as heat through turbulent fluid motion induced by breaking and 
through bottom friction and percolation. While the heat is of little 
concern to the coastal engineer, breaking is important since it affects 
both beaches and manmade shore structures. Thus, shore protection measures 
and coastal structure designs are dependent on the ability to predict wave 
forms and fluid motion beneath waves, and on the reliability of such 
predictions. Prediction methods generally have been based on simple waves 
where elementary mathematical functions can be used to describe wave motion, 
For some situations, simple mathematical formulas predict wave conditions 
well, but for other situations predictions may be unsatisfactory for 
engineering applications. Many theoretical concepts have evolved in the 
past two centuries for describing complex sea waves; however, complete 
agreement between theory and observation is not always found. 

In general, actual water-wave phenomena are complex and difficult 
to describe mathematically because of nonlinearities, three-dimensional 
characteristics and apparent random behavior. However, there are two 
classical theories, one developed by Airy (1845) and the other by Stokes 
(1880), that describe simple waves. The Airy and Stokes theories gener- 
ally predict wave behavior better where water depth relative to wavelength 
is not too small. For shallow water, a cnoidal wave theory often provides 
an acceptable approximation of simple waves. For very shallow water near 
the breaker zone, solitary wave theory satisfactorily predicts certain 
features of the wave behavior. These theories will be described according 
to their fundamental characteristics together with the mathematical equa- 
tions which describe wave behavior. Many other wave theories have been 
presented in the literature which, for some specific situations, may pre- 
dict wave behavior more satisfactorily than the theories presented here. 
These other theories are not included, since it is beyond the scope of 
this Manual to cover all theories. 

The most elementary wave theory, referred to as small-amplitude or 
linear wave theory, was developed by Airy (1845). It is of fundamental 
importance since it not only is easy to apply, but is reliable over a 
large segment of the whole wave regime. Mathematically, the Airy theory 
can be considered a first approximation of a complete theoretical descrip- 
tion of wave behavior. A more complete theoretical description of waves 
may be obtained as the sum of an infinite number of successive approxima- 
tions, where each additional term in the series is a correction to preced- 
ing terms. For some situations, waves are better described by these 
higher order theories which are usually referred to as finite amplitude 



2-2 



theories. The first finite amplitude theory, known as the trochoidal 
theory, was developed by Gerstner (1802). It is so called because the 
free surface or wave profile is a trochoid. This theory is mentioned 
only because of its classical interest. It is not recommended for appli- 
cation, since the water particle motion predicted is not that observed in 
nature. The trochoidal theory does, however, predict wave profiles quite 
accurately. Stokes (1880) developed a finite-amplitude theory which is 
more satisfactory than the trochodial theory. Only the second-order Stokes' 
equations will be presented, but the use of higher order approximations is 
sometimes justified for the solution of practical problems. 

For shallow-water regions, cnoidal wave theory, originally developed 
by Korteweg and De Vries (1895) , predicts rather well the waveform and 
associated motions for some conditions. However, cnoidal wave theory has 
received little attention with respect to actual application in the solu- 
tion of engineering problems. This may be due to the difficulties in 
making computations. Recently, the work involved in using cnoidal wave 
theory has been substantially reduced by introduction of graphical and 
tabular forms of functions. (Wiegel, 1960), (Masch and Wiegel, 1961.) 
Application of the theory is still quite involved. At the limit of cnoidal 
wave theory, certain aspects of wave behavior may be described satisfacto- 
rily by solitary wave theory. Unlike cnoidal wave theory, the solitary 
wave theory is easy to use since it reduces to functions which may be 
evaluated without recourse to special tables. 

Development of individual wave theories is omitted, and only the 
results are presented since the purpose is to present only that infor- 
mation which may be useful for the solution of practical engineering 
problems. Many publications are available such as Wiegel (1964), Kinsman 
(1965) , and Ippen (1966a), which cover in detail the development of some of 
the theories mentioned above as well as others. The mathematics used here 
generally will be restricted to elementary arithmetic and algebraic opera- 
tions. Emphasis is placed on selection of an appropriate theory in accord- 
ance with its application and limitations. 

Numerous example problems are provided to illustrate the theory 
involved and to provide some practice in using the appropriate equations 
or graphical and tabular functions. Some of the sample computations give 
more significant digits than are warranted for practical applications. 
For instance, a wave height could be determined to be 10.243 feet for 
certain conditions based on purely theoretical considerations. This 
accuracy is unwarranted because of the uncertainty in the basic data used 
and the assumption that the theory is representative of real waves. A 
practical estimate of the wave height given above would be 10 feet. When 
calculating real waves, the final answer should be rounded off. 

2.22 WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES 

Any adequate physical description of a water wave involves both its 
surface form and the fluid motion beneath the wave. A wave which can be 
described in simple mathematical terms is called a simple wave. Waves 



2-3 



which are difficult to describe in form or motion, and which may be com- 
prised of several components are termed complex waves. Sinusoidal or 
simple harmonic waves are examples of simple waves since their surface 
profile can be described by a single sine or cosine function. A wave is 
periodic if its motion and surface profile recur in equal intervals of 
time. A wave form which moves relative to a fluid is called a progressive 
wave; the direction in which it moves is termed the direction of wave 
propagation. If a wave form merely moves up and down at a fixed position, 
it is called a complete standing wave or a clapotis. A progressive wave 
is said to be a wave of permanent form if it is propagated without experi- 
encing any changes in free surface configuration. 

Water waves are considered osoiltatory or nearly oscillatory if the 
water particle motion is described by orbits, which are closed or nearly- 
closed for each wave period. Linear, or Airy, theory describes pure 
oscillatory waves. Most finite amplitude wave theories describe nearly 
oscillatory waves since the fluid is moved a small amount in the direction 
of wave advance by each successive wave. This motion is termed mass 
transport of the waves. When water particles advance with the wave, and 
do not return to their original position, the wave is called a wave of 
translation. A solitary wave is an example of a wave of translation. 

It is important to distinguish between various types of water waves 
that may be generated and propagated. One way to classify waves is by 
wave period T (the time for a wave to travel a distance of one wave 
length), or by the reciprocal of T, the wave frequency f. One illustra- 
tion of classification by period or frequency is given by Kinsman (1965) 
and shown in Figure 2-1. The figure shows the relative amount of energy 
contained in ocean waves having a particular frequency. Of primary concern 
are those waves referred to as gravity waves in Figure 2-1, having periods 
from 1 to 30 seconds. A narrower range of wave periods, from 5 to 15 
seconds, is usually more important in coastal engineering problems. Waves 
in this range are referred to as gravity waves since gravity is the 
principal restoring force; that is, the force due to gravity attempts to 
bring the fluid back to its equilibrium position. Figure 2-1 also shows 
that a large amount of the total wave energy is associated with waves 
classified as gravity waves; hence gravity waves are extremely important 
in dealing with the design of coastal and offshore structures. 

Gravity waves can be further separated into two states: 

(a) seas, when the waves are under the influence of wind in a 
generating area, and 

(b) swell, when the waves move out of the generating area and 
are no longer subjected to significant wind action. 

Seas are usually made up of steeper waves with shorter periods and 
lengths, and the surface appears much more confused than for swell. Swell 
behaves much like a free wave, i.e., free from the disturbing force that 
caused it, while seas consist to some extent of forced waves, i.e., waves 
on which the disturbing force is applied continuously. 

2-4 








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2-5 



Ocean waves are complex. Many aspects of fluid mechanics necessary 
to a complete discussion have only a minor influence on solving most 
coastal engineering problems. Thus, a simplified theory which omits most 
of the complicating factors is useful. The assumptions made in developing 
the simple theory should be understood, because not all of the assumptions 
are justified in all problems. When an assumption is not valid in a 
particular problem, a more complete theory should be employed. 

The most restrictive of common assumptions is that waves are small 
perturbations on the surface of a fluid which is otherwise at rest. This 
leads to a wave theory which is variously called, small-amplitude theory, 
linear theory, or Airy theory. The small-amplitude theory provides in- 
sight for all periodic wave behavior and a description of the periodic flow 
adequate for most practical problems. This theory is unable to account for 
mass transport due to waves (Section 2.253 Mass Transport Velocity), or the 
fact that wave crests depart further from the mean water level than do the 
troughs. A more general theory, usually called the finite amplitude, or 
nonlinear wave theory is required to account for these phenomena as well as 
most interactions between waves and other flows. The nonlinear wave theory 
also permits a more accurate evaluation of some wave properties than can be 
obtained with linear theory. 

Several assumptions, commonly made in developing a simple wave theory 
are listed below. 

(a) The fluid is homogeneous and incompressible; therefore, the 
density p is a constant. 

(b) Surface tension can be neglected. 

(c) Coriolis effect can be neglected. 

(d) Pressure at the free surface is uniform and constant. 

(e) The fluid is ideal or inviscid (lacks viscosity) . 

(f) The particular wave being considered does not interact with 
any other water motions. 

(g) The bed is a horizontal, fixed, impermeable boundary which 
implies that the vertical velocity at the bed is zero. 

(h) The wave amplitude is small and wave form is invariant in 
time and space. 

(i) Waves are plane or long crested (two-dimensional) . 

The first three are acceptable for virtually all coastal engineering 
problems. It will be necessary to relax assumptions (d) , (e), and (f) 
for some specialized problems not considered in this Manual. Relaxing 
the three final assumptions is essential in many problems, and is con- 
sidered later in this chapter. 

2-6 



In applying assumption (g) to waves in water of varying depth encoun- 
tered when waves approach a beach the local depth is usually used. This 
can be rigorously justified, but not without difficulty, for most practical 
cases in which the bottom slope is flatter than about 1 on 10. A progres- 
sive wave moving into shallow water will change its shape significantly. 
Effects due to viscosity and vertical velocity on a permeable bottom may 
be measurable in some situations, but these effects can be neglected in 
most engineering problems. 

2.23 ELEMENTARY PROGRESSIVE WAVE THEORY (Small -Amplitude Wave Theory) 

The most fundamental description of a simple sinusoidal oscillatory 
wave is by its length L (the horizontal distance between corresponding 
points on two successive waves) ; height H (the vertical distance to its 
crest from the preceding trough) ; period T (the time for two successive 
crests to pass a given point) ; and depth d (the distance from the bed 
to the Stillwater level). (See Appendix B for a list of common symbols.) 

Figure 2-2 shows a two-dimensional simple progressive wave propagating 
in the positive x-direction. The symbols used here are presented in the 
figure. The symbol n denotes the displacement of the water surface 
relative to the Stillwater level (SWL) and is a function of x and time. 
At the wave crest, t\ is equal to the amplitude of the wave a, or one- 
half of the wave height. 

Small -amplitude wave theory and some finite-amplitude wave theories 
can be developed by introduction of a velocity potential <t)(x, z, t) . Hori- 
zontal and vertical components of the water particle velocities are defined 
at a point (x, z) in the fluid as u = 8<(i/9x and w = 3(|)/3z. The velocity 
potential, Laplace's equation, and Bernoulli's dynamic equation together 
with the appropriate boundary conditions provide the necessary information 
needed in deriving the small-amplitude wave formulas. Such a development 
has been shown by Lamb (1932), Eagleson and Dean (See Ippen 1966b), and 
others . 

2.231 Wave Celerity, Length and Period . The speed at which a wave form 
propagates is termed the phase velocity or wave celerity, C. Since the 
distance traveled by a wave during one wave period is equal to one wave- 
length, the wave celerity can be related to the wave period and length by 

C = ^ (2-1) 

An expression relating the wave celerity to the wavelength and water depth 
is given by 



C = 




(2-2) 



2-7 



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2-8 



From Equation 2-1, it is seen that 2-2 can be written as 



gT , /27rd 

C = — tanh 

2iT \ L > 



(2-3) 



The values 2Tr/L and 2Tr/T are called the wave number k and wave angular 
frequency w, respectively. From Equations 2-1 and 2-3 an expression 
for wavelength as a function of depth and wave period may be obtained. 



L = 



2-n 



tanh 




(2-4) 



Use of Equation 2-4 involves some difficulty since the unknown L, appears 
on both sides of the equation. Tabulated values in Appendix C may be used 
to simplify the solution of Equation 2-4. 

Gravity waves may also be classified by the depth of water in which 
they travel. Classification is made according to the magnitude of d/L 
and the resulting limiting values taken by the function tanh(27rd/L) . 
Classifications are: 



Classification 


d/L 


27rd/L 


tanh (27rd/L) 


Deep Water 


> 1/2 


> TT 


X 1 


Transitional 


1/25 to 1/2 


1/4 to TT 


tanh (27rd/L) 


Shallow Water 


< 1/25 


< 1/4 


X 2jrd/L 



In deep water, tanh(2TTd/L) approaches unity and Equations 2-2 and 2-3 
reduce to 



and 



"gLT L, 



^o V 27r T 



° lit 



(2-5) 



(2-6) 



Although deep water actually occurs at infinite depth, tanh(2ird/L) , 
for most practical purposes, approaches unity at a much smaller d/L. For 
a relative depth of 1/2 (that is, when the depth is one-half the wavelength), 
tanh(2Trd/L) = 0.9964. 

Thus, when the relative depth d/L, is greater than 1/2, the wave 
characteristics are virtually independent of depth. Deepwater conditions 



are indicated by the subscript 



as in 



and Cq. The period T, 



remains constant and independent of depth for oscillatory waves; hence the 



2-9 



subscript is omitted. (Ippen, 1966b, pp 21-240 If units of feet and 
seconds are specified, the constant g/2Tr is equal to 5.12 ft/sec^ and 



and 



C = ^ = 5.12 T (ft/sec) , (2-7) 

" 2-n 



L^ = ^— = 5.12 T^ (ft). (2-8) 



If Equation 2-7 is used to compute wave celerity when the relative depth 
is d/L = 0.25, the resulting error will be about 9 percent. It is evi- 
dent that a relative depth of 0.5 is a satisfactory boundary separating 
deepwater waves from waves in water of transitional depth. If a wave is 
traveling in transitional depths. Equations 2-2 and 2-3 must be used with- 
out simplification. Care should be exercised to use Equations 2-2 and 2-3 
when necessary, that is, when the relative depth is between 1/2 and 1/25. 

When the relative water depth becomes shallow, i.e., 2TTd/L < 1/4 or 
d/L < 1/25, Equation 2-2 can be simplified to 

C = yf^. (2-9) 

This relation, attributed to Lagrange, is of importance when dealing with 
long-period waves, often referred to as long waves. Thus, when a wave 
travels in shallow water, wave celerity depends only on water depth. 

2.232 The Sinusoidal Wave Profile . The equation describing the free 
surface as a function of time t, and horizontal distance x, for a 
simple sinusoidal wave can be shown to be 

17 = a cos I — 1 = — cos I — -— ) , (2-10) 



where n is the elevation of the water surface relative to Stillwater 
level, and H/2 is one-half the wave height equal to the wave amplitude 
a. This expression represents a periodic, sinusoidal, progressive wave 
traveling in the positive x-direction. For a wave moving in the negative 
x-direction, one need only replace the minus sign before 2TTt/T with a 
plus sign. When (2Trx/L - 2Trt/T) equals 0, Tr/2, tt, 37v/2, the corresponding 
values of n are H/2, 0, - H/2, and 0, respectively. 

2.233 Some Useful Functions . It can be shown by dividing Equation 2-3 by 
Equation 2-6, and by dividing Equation 2-4 by Equation 2-8 that 





(2-11) 



2-10 



If both sides of Equation 2-11 are multiplied by d/L, it becomes: 

d d , /27rd\ 

— = - tanh (2-12) 

The term d/L^ has been tabulated as a function of d/L by Wiegel (1954), 
and is presented in Appendix C on Table C-1. Table C-2 includes d/L as 
a function of d/L^ iri addition to other useful functions such as 2ird/L 
and tanh(2Trd/L) . These functions simplify the solution of wave problems 
described by the linear theory. 

An example problem illustrating the use of linear wave theory and 
the tables in Appendix C follows: 



********* *** * EXAMPLE PROBLEM 



************** 



GIVEN : A wave with a period of T = 10 seconds is propagated shoreward 
over a uniformly sloping shelf from a depth of d = 600 feet to a depth 
of d = 10 feet. 

FIND : The wave celerities C and lengths L corresponding to depths of 
d = 600 feet and d = 10 feet. 

SOLUTION: 



Using Equation 2-8, 



L^ = 5.12 T^ = 5.12 (10)^ = 512 feet. 



For d = 600 feet 

d 600 



Lo 512 



1.1719. 



From Table C-1 it is seen that for values of 

d 

->1.0 



K L' 



therefore 



fa. 



L - L^ = 512 feet (deepwater wave, since — > — 



2-1 



By Equation 2-1 



L 
T 



512 
T 



For d = 10 feet 



512 
C = — = 51.2 ft/sec. 



Z; ~ 512 



0.0195. 



Entering Table C-1 with d/L^ it is found that. 



hence 



10 



0.05692 



- = 0.05692. 
L 



. r I . . , , , 1 d l\ 

= 176 reet [transitional depth, since — < — <—), 

\ Ad i-f A j 



c = 



176 
10 



= 17.6 ft/sec. 



2.234 Local Fluid Velocities and Accelerations . In wave force studies, 
it is often desirable to know the local fluid velocities and accelerations 
for various values of z and t during the passage of a wave. The 
horizontal component u, and the vertical component w, of the local 
fluid velocity are given by; 



u = 



H gT cosh[27r(z + d)/L] 



2 L 



w = — 



cosh (27rd/L) 



H gT sinh[27r(z + d)/L] 
2 L cosh (27rd/L) 



cos 



sin 



27rx 


lut 


L 


T 


2-nx 


27rt 



(2-13) 



(2-14) 



These equations express the local fluid velocity components any distance 
(z + d) above the bottom. The velocities are harmonic in both x and t. 
For a given value of the phase angle 6 = (2irx/L - 2iTt/T) , the hyperbolic 
functions, aosh and sinhy as functions of z result in an approximate 
exponential decay of the magnitude of velocity components with increasing 
distance below the free surface. The maximum positive horizontal velocity 



2-12 



occurs when 9 = 0, 2tt, etc., while the maximum horizontal velocity in the 
negative direction occurs when 9 = tt. Sit, etc. On the other hand the 
maximum positive vertical velocity occurs when 9 = tt/2, 5tt/2, etc., and 
the maximum vertical velocity in the negative direction occurs when 
e = 377/2, 777/2, etc. (See Figure 2-3.) 

The local fluid particle accelerations are obtained from Equations 
2-13 and 2-14 by differentiating each equation with respect to t. Thus, 

gTTH cosh[27r(z + d)/L] . , _..^ „„. , ._ ^_. 
"- = "" IT cosh (277d/L) "" '-^ - ^^- ^'"''^ 



277X 


27rt 


L 


T 


27rx 


2;rt' 


L 


T 



gTTH sinh[277(z + d)/L] , -,_ .„., ,_ _, 
'^^ = " IT cosh (2.d/L) '''' [-^ - ^'- ^'"''^ 

Positive and negative values of the horizontal and vertical fluid 
accelerations for various values of 9 = 277x/L - 277t/T are shown in 
Figure 2-3. 

The following problem will illustrate the computations required to 
determine local fluid velocities and accelerations resulting from wave 
motions. 



************** EXAMPLE PROBLEM 



************** 



GIVEN : A wave with a period of T = 8 seconds, in a water depth of 
d = 50 feet, and a height of H = 18 feet. 

FIND : The local horizontal and vertical velocities, u and w, and 
accelerations a^ and a.^ at a depth d = 15 feet below the SWL 
when 9 = 277x/L -277t/T = 7t/3 (60 degrees). 

SOLUTION : Calculate 

L^ = 5.12 T^ = 5.12(8)^ = 328 feet , 



50 

= 0.1526 . 



L, 328 



From Table C-1 in Appendix C for a value of 



d 

— = 0.1526 , 
L 



d , 27rd 

- « 0.1854; cosh = 1.759 , 

L L 



2-13 




I/) 

c 
o 



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<U 
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< 

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C 

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o 



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o 
o 



to 
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CM 

0) 

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2-14 



hence 



L = 



50 



= 270 feet 



0.1854 
Evaluation of the constant terms in Equations 2-13 through 2-16 gives 



HgT 
2L 

gTTH 



cosh (27rd/L) 

1 
cosh (27rd/L) 



18 (32.2) (8) 
2(270) (1.758) 

18 (32.2) (tt) 



(270) (1.758) 



4.88, 



= 3.84. 



Substitution into Equation 2-13 gives 



u = 4.88 cosh 
From Table C-1 find 



27:(50- 15) 



270 



[cos 60°] = 4.88 [cosh (0.8145)] (0.500). 



lird 



= 0.8145 , 



and by interpolation 



and 



Therefore 



u = 
w = 



"x = 



a = 

z 



cosh (0.8145) = 1.3503, 
sinh (0.8145) = 0.9074. 

4.88 (1.3503) (0.500) = 3.29 ft/sec , 
4.88 (0.9074) (0.866) = 3.83 ft/sec , 
3.84 (1.3503) (0.866) - 4.49 ft/sec^ 
-3.84 (0.9074) (0.500) = - 1.74 ft/sec^, 



Figure 2-3, a sketch of the local fluid motion, indicates that the 
fluid under the crest moves in the direction of wave propagation and 
returns during passage of the trough. Linear theory does not predict any 
mass transport; hence the sketch shows only an oscillatory fluid motion. 

************************************* 

2.235 Water Particle Displacements . Another important aspect of linear 
wave mechanics deals with the displacements of individual water particles 
within the wave. Water particles generally move in elliptical paths in 
shallow or transitional water and in circular paths in deep water. If the 



2-15 



mean particle position is considered to be at the center of the ellipse or 
circle, then vertical particle displacement with respect to the mean 
position cannot exceed one-half the wave height. Thus, since the wave 
height is assigned to be small, the displacement of any fluid particle from 
its mean position is small. Integration of Equations 2-13 and 2-14 gives 
the horizontal and vertical particle displacement from the mean position, 
respectively. (See Figure 2-4.) 



Thus, 



I = - 



HgT^ cosh[27r(z + d)/L] 



47rL 



cosh (27rd/L) 



sin 




(2-17) 



HgT^ smh[27:(z + d)/L] /27rx l-nt^ 

f = + —2 ; ; cos 

^ 477 L cosh (277 d/L) \ L T ; 



(2-18) 



The above equations can be simplified by using the relationship 



Thus, 




2ire , 277d 
— ^ tanh . 



H cosh [277(z + d)/L] . /27rx 277t 

— ; sin I — — — 

2 sinh(277d/L) \ L T 



(2-19) 



H sinh [277(z + d)/L] /277X 277t \ 

— ; cos I — - I. 

2 sinh (277d/L) \ L T / 



(2-20) 



Writing Equations 2-19 and 2-20 in the following forms: 



sin 



cos 




277X 277t 



27rt 
T 



i, sinh (277d/L) 

a cosh [277(z+d)/L. 

j sinh (27rd/L) 

a sinh[277(z+d)/L]_ 



and adding, gives: 



— + ^r = 1, 



(2-21) 



in which 



A = 



H cosh[277(z+d)/L] 
2 sinh (277d/L) 



(2-22) 



H sinh[277(z+d)/L] 
^ " 2 sinh(277d/L) ' 



(2-23) 



2-16 




(D 

> 
CO 



<D 

CI, 
(D 

Q 

TD 
C 



o 



cfl 

o 

c 
o 



o 

a. 

c 

0) 



S 
o 
u 

U) 

c 

0) 
E 
(D 
O 
c« 

i-H 

cx 

Q 

(U 

I— I 

o 



ni 
U 



3 

•H 



2-17 



Equation 2-21 is the equation of an ellipse with a major (horizontal) 
semiaxis equal to A, and a minor (vertical) semiaxis equal to B. The 
lengths of A and B are measures of the horizontal and vertical dis- 
placements of the water particles. Thus, the water particles are predicted 
to move in closed orbits by linear wave theory: i.e., each particle returns 
to its initial position after each wave cycle. Morison and Crooke (1953), 
compared laboratory measurements of particle orbits with wave theory and 
found, as had others, that particle orbits were not completely closed. This 
difference between linear theory and observations is due to the mass trans- 
port phenomenon which is discussed in a subsequent section. 

Examination of Equations 2-22 and 2-23 shows that for deepwater 
conditions A and B are equal and particle paths are circular. The 
equations become 

A=B = -e'^^/L for->-. (2-24) 

2 L 2 

For shallow-water conditions, the equations become 



B 



H 


L 




2 


27rd 


c ^ 1 

for - < — 


H 


z + d 


L 25 


2 


d 





(2-25) 



Thus, in deep water, the water particle orbits are circular. The more 
shallow the water, the flatter the ellipse. The amplitude of the water 
particle displacement decreases exponentially with depth and in deepwater 
regions becomes small relative to the wave height at a depth equal to 
one-half the wavelength below the free surface, i.e., when z = - Lo/2. 
This is illustrated in Figure 2-4. For shallow regions, horizontal 
particle displacement near the bottom can be large. In fact, this is 
apparent in offshore regions seaward of the breaker zone where wave action 
and turbulence lift bottom sediments into suspension. 

The vertical displacement of water particles varies from a minimum of 
zero at the bottom to a maximum equal to one-half the wave height at the 
surface. 

************** EXAMPLE PROBLEM ************** 

PROVE : 

/-„^ /27rV 27rg 

(a) = _S tanh 




(b) 



T / L 



jtH cosh[27r(z+d)/L] /27r: 

u = — ; ; cos 

T sinh(27rd/L) \ L 



2-18 



SOLUTION : 

(a) Equation 2-3, 

Equation 2-1, 



2ir \ L , 



T 



Therefore, equating 2-1 and 2-3, 



T 2n \ L r 



and multiplying both sides by (2tt)2/lt 



Hence, 



(2i:y L _ (2nr gT , /27:d^ 
LT T = IF ^ '^"MX; 



i2ny 27rg ^ Ayrd^ 



(bj Equation 2-13 may be written 

^ ^ gTH cosh[27r(z+d)/L] /2nx _ 27Tt\ 

2L cosh (27rd/L) """M L ~ T J 

^ 1 gH cosh[27r(z + d)/L] hnx 27rt^ 

C 2 cosh(27rd/L) ^""llT ~ F; 



since 



T _ 1 
L ~ C 



Since 



C = C .anh M 
2n \ L J 



^ ^ ^ 1_ cosh[27r(z + d)/L] /27rx _ 2n^ 

T tanh(27rd/L) cosh (27rd/L) ^°' I L ~T"y 



2-19 



and since, 

/27rd\ _ sinh (27rd/L) 
y L y ~ cosh (27rd/L) ' 

therefore, 

■nH cosh [27r(z+d)/L] (inx 27rt\ 

U = — -— ; COS I — I . 

T sinh (27rd/L) \ L T / 
************************************* 

************** EXAMPLE PROBLEM ************** 

GIVEN : A wave in a depth of d = 40 feet, height of H = 10 feet, and a 
period of T = 10 seconds. The corresponding deepwater wave height is 
Hq = 10.45 feet. 

FIND : 

(a) The horizontal and vertical displacement of a water particle from 
its mean position when z = 0, and when z = - d. 

(b) The maximum water particle displacement at a depth d = 25 feet when 
the wave is in infinitely deep water. 

(c) For the deepwater conditions of (b) above, show that the particle 
displacements are small relative to the wave height when z = - Lo/2. 

SOLUTION : 

(a) L^ = 5.12 T^ = 5.12(10)^ = 512 feet , 

d 40 

0.0781 . 



Lc 512 



From Appendix C, Table C-1 



sinh ( — 1= 0.8394 , 



1 /27rd\ 
tanh I 1= 0.6430 . 



When z = 0, Equation 2-22 reduces to 

H 1 



A = 



2 tanh(27rd/L) ' 
2-20 



and Equation 2-23 reduces to 

H 
B = - 

2 



Thus, 



10 1 

A = = 7.78 feet , 

2 (0.6430) 



H 10 

B = - = — = 5.0 feet 
2 2 



When z = - d, 

A = = = 5 96 feet 

2sinh(27rd/L) 2(0.8394) 

and, B = 0. 

(b) With Uo = 10-45 feet, and z = - 25, evaluate the exponent of e 
for use in Equation 2-24, noting that L = L^, 

/27rz\ 27r(- 25) 

= — ^ = - 0.307 , 

\L I 512 

thus 



e-0.307 = 0.736 . 



Therefore, 



H 27:2/ 10.45 

A=B= — e ^^ = (0.736) = 3.85 feet . 

2 2 

The maximum displacement or diameter of the orbit circle would 
be 2(3.85) = 7.70 feet. 

r ^ ^o "512 

Cc) z = = = - 256 feet , 

2 2 

27rz 2-n(- 256) 

-— = = - 3.142 . 

L 512 

Using Table C-4 of Appendix C, 

g-3.142 ^ 0.043. 



H^ 2mL 10.45 

~ e f^ = -^^— (0.043) = 0.225 feet . 

2-21 



A = B 

2 2 



Thus the maximum displacement of the particle is 0.45 feet which is small 
when compared with the deepwater height, H^ = 10.45 feet. 

************************************* 

2.236 Subsurface Pressure . Subsurface pressure under a wave is the 
summation of two contributing components, dynamic and static pressures, 
and is given by 

coshl27r(z + d)/L] H /27rx 27rt\ 

P = '^ cosh (2.d/L) 2 ^°^ \y~^l ' '^" ^ ^^ ' ^'-'^^ 

where p' is the total or absolute pressure, p^^ is the atmospheric 
pressure and p = w/g is the mass density of water (for saltwater, 
p = 2.0 lbs sec2/ft'+= 2.0 slugs/ft^; for fresh water, p = 1.94 slugs/ft^). 
The first term of Equation 2-26 represents a dynamic component due to 
acceleration, while the second term is the static component of pressure. 
For convenience, the pressure is usually taken as the gage pressure 
defined as 

cosh[27r(z+d)/L] H /27rx 27rt\ 

^=^'-^a = P% eosh(2.d/L) 2 '^"^ I^IT "^ T"]" '^' ' ^'"''^ 

Equation 2-27 can be written as 

cosh[27r(z+d)/L] 

p = Pgr? Pgz , (2-28) 

cosh (27Td/L) 



since 



The ratio 



H /27rx 27rtl 

7? = — cos I — 

2 \ L T ; 



cosh[27r(z+d)/L] 



K = -^-^ . (2-29) 

^ cosh(27rd/L) 

is termed the pressure response factor. Hence, Equation 2-28 can be 
written as 

p = pg(rjK -z) . (2-30) 



2-22 



The pressure response factor K, for the pressure at the bottom when 
z = - d, 

1 

K, = K = — 1 .^ ,,.-> , (2-31) 

2 cosh (ZTrd/L) ' 

is tabulated as a function of d/L^ and d/L in Tables C-1 and C-2 of 
Appendix C. 

It is often necessary to determine the height of surface waves based 
on subsurface measurements of pressure. For this purpose it is convenient 
to rewrite Equation 2-30 as, 

_ N(p+pgz) 
r? - , (2-32) 

where z is the depth below the SWL of the pressure gage, and N is a 
correction factor equal to unity if the linear theory applies. Several 
empirical studies have found N to be a function of period, depth, wave 
amplitude and other factors. In general, N decreases with decreasing 
period, being greater than 1.0 for long-period waves and less than 1.0 for 
short-period waves. 

A complete discussion of the interpretation of pressure gage wave 
records is beyond the scope of this Manual. For a more detailed discussion 
of the variation of N with wave parameters, the reader is referred to 
Draper (1957), Grace (1970), and Esteva and Harris (1971). 

************** EXAMPLE PROBLEM ************** 

GIVEN : An average maximum pressure of p = 2590 lbs/ft^ is measured by a 
subsurface pressure gage located 2 feet above the bed in water at d = 40 
feet. The average frequency f = 0.0666 cycles per second. 

FIND : The height of the wave H assuming that linear theory applies and 
the average frequency corresponds to the average wave amplitude. 



SOLUTION: 



1 1 

T = — = ^ _^^^ ~ 15 seconds 

f (0.0666) 



L^ = 5.12 T^ = 5.12(15)^ = 1152 feet 



K 1152 



40 

0.0347 



2-23 



From Table C-1 o£ Appendix C, entering with d/L^, 

d 



hence 



and 



0.07712 
L 



40 

L = ; = 519 feet , 

(0.07712) 



cosh I 1= 1.1197 



Therefore, from Equation 2-29 

cosh[27r(z + d)/L] cosh [27r(- 38 + 40)/519] 1.0003 
K = . .„ .,^. = = = 0.8934 . 



z 



cosh(27rd/L) 1.1197 1.1197 



Since n = a = H/2 when the pressure is maximiom (under the wave crest) , 
and N = 1.0 since linear theory is assumed valid. 



H N(p+pgz) 1.0 [2590 + (64.4) (-38)] 

- = ,, = = 2.47 feet 

2 pgK (64.4) (0.8934) 



Therefore, 



H = 2 (2.47) ^ 5 feet 



Note that the tabulated value of K in Appendix C, Table C-1, could not 
be used since the pressure was not measured at the bottom. 



2.237 Velocity of a Wave Group . The speed with which a group of waves 
or a wave train travels is generally not identical to the speed with which 
individual waves within the group travel. The group speed is termed the 
group velocity, C_; the individual wave speed is the phase velocity or 
wave celerity given by Equations 2-2 or 2-3. For waves propagating in 
deep or transitional water with gravity as the primary restoring force, 
the group velocity will be less than the phase velocity. (For those waves 
propagated primarily under the influence of surface tension, i.e., capil- 
lary waves, the group velocity may exceed the velocity of an individual 
wave . ) 

The concept of group velocity can be described by considering the 
interaction of two sinusoidal wave trains moving in the same direction 



2-24 



with slightly different wavelengths and periods, 
water surface is given by: 



The equation of the 



H /27rx 

+ n, = — cos I 

' 2 \ L. 



27rt\ 


H 


/27rx 


2iit 


-. 


+ — cos 
2 


u. 


T. 



-— , (2-33) 



where n, and r) are the contributions of each of the two components. 

They may be summed since superposition of solutions is permissible when 
linear wave theory is used. For simplicity, the heights of both wave 
components have been assumed equal. Since the wavelengths of the two 
component waves, L-,^ and L2, have been assumed slightly different, for 
some values of x at a given time, the two components will be in phase 
and the wave height observed will be 2H; for some other values of x, 
the two waves will be completely out of phase and the resultant wave 
height will be zero. The surface profile made up of the stun of the two 
sinusoidal waves is given by Equation 2-33 and is shown in Figure 2-5. 
The waves shown on Figure 2-5 appear to be traveling in groups described 
by the equation of the envelope curves: 



^envelope 



= ± H cos 






^2 - T, 



(2-34) 



It is the speed of these groups, i.e. the velocity of propagation of 
the envelope curves, that represents the group velocity. The limiting 
speed of the wave groups as they become large, i.e., as the wavelength, 
L^, approaches L2 and consequently the wave period T-^ approaches T^ 
is the group velocity and can be shown to be equal to: 



s = 



1 h 

2 T 



1 + 



47Td/L 



sinh (47rd/L) 



= nC 



(2-35) 



where 



1 

n = — 
2 



1 + 



47rd/L 



sinh (47rd/L) 
In deep water, the term (47rd/L)/sinh(4Trd/L) is approximately zero and, 



1 L^ 1 
^g " 2 Y ^ 2 ^° (deep water) 



(2-36) 



or the group velocity is one-half the phase velocity. 
sinh(47rd/L) ^ 4TTd/L and, 



In shallow water. 



C = - = C =« 

S T 



gd (shallow water) 



(2-37) 



2-25 



v^'n^^Vz 



^envelope 




-0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 



X /J-2J-1 \ t /Jj_~Xl \ 
~2 \~r r )~ ~2 \'~j — J — J ( offer Kinsman, 1965) 



Figure 2-5. Formation of Wave Groups by the Addition of Two Sinusoids 
Having Different Periods 



2-26 



hence the group and phase velocities are equal. Thus in shallow water, 
because wave celerity is fully determined by the depth, all component waves 
in a wave train will travel at the s^nie speed precluding the alternate 
reinforcing and cancelling of components. In deep and transitional water, 
wave celerity depends on the wavelength; hence slightly longer waves travel 
slightly faster, and produce the small phase differences resulting in wave 
groups. These waves are said to be dispersive or propagating in a 
dispersive mediioriy i.e. in a medium where their celerity is dependent on 
wavelength . 

Outside of shallow water, the phase velocity of gravity waves is 
greater than the group velocity, and an observer moving along with a group 
of waves at the group velocity will see waves that originate at the rear 
of the group move forward through the group traveling at the phase velocity, 
and disappear at the front of the wave group. 

Group velocity is important because it is with this velocity that wave 
energy is propagated. 

Although mathematically, the group velocity can be shown rigorously 
from the interference of two or more waves (Lamb, 1932), the physical 
significance is not as obvious as it is in the method based on the con- 
sideration of wave energy. Therefore an additional explanation of group 
velocity is provided on wave energy and energy transmission. 

2.238 Wave Energy and Power . The total energy of a wave system is the 
sum of its kinetic energy and its potential energy. The kinetic energy is 
that part of the total energy due to water particle velocities associated 
with wave motion. Potential energy is that part of the energy resulting 
from part of the fluid mass being above the trough - the wave crest. 
According to the Airy theory, if the potential energy is determined 
relative to mean water level, and all waves are propagated in the same 
direction, potential and kinetic energy components are equal, and the 
total wave energy in one wavelength per unit crest width is given by 

Subscripts k and p refer to kinetic and potential energies. Total 
average wave energy per unit surface area, termed the specific energy or 
energy density, is given by 

E pgH* 
E = - = ^ . (2-39) 

Wave energy flux is the rate at which energy is transmitted in the 
direction of wave propagation across a vertical plane perpendicular to the 
direction of wave advance and extending down the entire depth. The average 



2-27 



energy flux per unit wave crest width transmitted across a plane perpen- 
dicular to wave advance is 

P = InC = EC . (2-40) 

g 

Energy flux F is frequently called wave power and 



47rd/L 
1 + 



sink (47rd/L) 



If a plane is taken other than perpendicular to the direction of wave 
advance F = F C sin (J), where (j) is the angle between the plane across 
which the energy is being transmitted and the direction of wave advance. 

For deep and shallow water. Equation 2-40 becomes 



P^ = - E^C (deep water). (2-41) 

o 2 " " *^ 



P = EC^ = EC (shallow water) . (2-42) 

An energy balance for a region through which waves are passing will 
reveal, that for steady state, the amount of energy entering the region 
will equal the amount leaving the region provided no energy is added or 
removed from the system. Therefore, when the waves are moving so that 
their crests are parallel to the bottom contours. 



Eo 


"oC, 




EnC 
1 




"o 




2 ' 


1 


_ 




_ 



or, since 

1 C^ = EnC . (2-43) 



2 



o o 



When the wave crests are not parallel to the bottom contours, some parts 
of the wave will be traveling at different speeds, the wave will be 
refracted and Equation 2-43 does not apply. (See Section 2.3. WAVE 
REFRACTION.) 

The following problem illustrates some basic principles of wave energy 
and energy flux. 



2-28 



************** EXAMPLE PROBLEM ************** 

GIVEN : A deepwater oscillatory wave with a wavelength of L^ = 512 feet, 
a height of H^ = 5 feet and a celerity of C^^ = 51.2 ft/sec, moving 
shoreward with its crest parallel to the depth contours. Any effects 
due to reflection from the beach are negligible. 

FIND : 

(a) Derive a relationship between the wave height in any depth of water 
and the wave height in deep water, assuming that wave energy per 
unit crest width is conserved as a wave moves from deep water into 
shoaling water. 

(b) Calculate the wave height for the given wave when the depth is 10 
feet. 

(c) Determine the rate at which energy per unit crest width is trans- 
ported toward the shoreline and the total energy per unit width 
delivered to the shore in 1 hour by the given waves . 

SOLUTION : 

(a) Since the wave crests are parallel to the bottom contours, refraction 
does not occur, therefore H^ = H^. (See Section 2.3. WAVE 
REFRACTION.) 

From Equation 2-43, 











1 - 


Co 


= nEC 


The 


expressions 


for 


Eo 


and 


E 


are, 

pgH;' 

8 ' 


and 








E 




PgH^ 



8 

where H^ represents the wave height in deep water if the wave 
were not refracted. 

Substituting into the above equation gives. 



2-29 



Therefore, 



1-1 



2 n C 



and since from Equations 2-3 and 2-6 

C , /27Td^ 

— = tanh 



and from Equation 2-35 where 

47:d/L 



1 + 



sink (477 d/L) 



H 



1 



1 



tanh (27rd/L) 



1 + 



(47rd/L) 



sinh (4rrd/L) 



K. 



(2-44) 



where Kg or H/H^ is termed the shoaling coefficient. Values 
of H/Hq as a function of d/Lo and d/L have been tabulated 
in Tables C-1 and C-2 of Appendix C. 

(b) For the given wave, d/L^ = 10/512 = 0.01953. Either from Table 
C-1 or from an evaluation of Equation 2-44 above. 



H 
H 



7 = 1.233 



Therefore , 



H = 1.233(5) = 6.165 ft. 



(c) The rate at which energy is being transported toward shore is the 
wave energy flux. 



P = - E^C = nEC . 
2 o o 

Since it is easier to evaluate the energy flux in deep water, 
the left side of the above equation will be used. 

- 1 - 1 PgCH^)' 51.2 1 64(5)^ 

P = - E^C = = 51.2 , 

2 " " 2 8 2 8 



P = 5120 



ft-lb 



sec 



per ft . of wave crest , 



P = 



5120 
550 



9.31 horsepower per ft. of wave crest . 



2-30 



This represents the expenditure of 

ft-lb sec , . ,, 

5120 X 3600 — = 18.55 X 10* ft-lbs , 

sec hr 

of energy each hour on each foot of beach. 

************************************* 

The mean rate of energy transmission associated with waves propagating 
into an area of calm water provides a better physical description of the 
concept of group velocity. An excellent treatment of this subject is given 
by Sverdrup and Munk (1947) and is repeated here. 

Quoting from the Beach Erosion Board Technical 
Report No. 2, (1942) : "As the first wave in the group 
advances one wave length, its form induces correspond- 
ing velocities in the previously undisturbed water and 
the kinetic energy corresponding to those velocities 
must be drawn from the energy flowing ahead with the 
form. If there is equipartition of energy in the wave, 
half of the potential energy which advanced with the 
wave must be given over to the kinetic form and the 
wave loses height. Advancing another wave length 
another half of the potential energy is used to supply 
kinetic energy to the undisturbed liquid. The process 
continues until the first wave is too small to identify. 
The second, third, and subsequent waves move into water 
already disturbed and the rate at which they lose height 
is less than for the first wave. At the rear of the 
group, the potential energy might be imagined as moving 
ahead, leaving a flat surface and half of the total 
energy behind as kinetic energy. But the velocity 
pattern is such that flow converges toward one section 
thus developing a crest and diverges from another 
section forming a trough. Thus the kinetic energy is 
converted into potential and a wave develops in the 
rear of the group." 

This concept can be interpreted in a quantitative 
manner, by taking the following example from R. Gatewood 
(Gaillard 1904, p. 50). Suppose that in a very long 
trough containing water originally at rest, a plunger 
at one end is suddenly set into harmonic motion and 
starts generating waves by periodically imparting an 
energy E/2 to the water. After a time interval of n 
periods there are n waves present. Let m be the posi- 
tion of a particular wave in this group such that m=l 
refers to the wave which has just been generated by 
the plunger, m={n+\)/2 to the center wave, and m=n to 
the wave furthest advanced. Let the waves travel with 
constant velocity C, and neglect friction. 

2-31 



After the first complete stroke one wave will be 
present and its energy is E/2. One period later this 
wave has advanced one wave length but has left one- 
half of its energy or E/4 behind. It now occupies a 
previously undisturbed area to which it has brought 
energy E/4. In the meantime, a second wave has been 
generated, occupying the position next to the plunger 
where E/4 was left behind by the first wave. The 
energy of this second wave equals E/4 + E/2 = 3E/4. 
Repeated applications of this reasoning lead to the 
results shown in Table 2-1. 

The series number n gives the total number of 
waves present and equals the time in periods since 
the first wave entered the area of calm; the wave 
number m gives the position of the wave measured from 
the plunger and equals the distance from the plunger 
expressed in wave lengths. In any series, n, the 
deviation of the energy from the value E/2 is 
symmetrical about the center wave. Relative to the 
center wave all waves nearer the plunger show an 
excess of energy and all waves beyond the center wave 
show a deficit. For any two waves at equal distances 
from the center wave the excess equals the deficiency. 
In every series, n, the energy first decreases slowly 
with increasing distance from the plunger, but in the 
vicinity of the center wave it decreases rapidly. 
Thus, there develops an "energy front" which advances 
with the speed of the central part of the wave system, 
that is, with half the wave velocity. 



According to the last line in Table 2-1 a definite 
pattern develops after a few strokes: the wave closest 
to the plunger has an energy E(2"-l)/2" which approaches 
the full amount E, the center wave has an energy E/2, 
and the wave which has traveled the greatest distance 
has very little energy (E/2^) . 

Table 2-1. Distribution of Wave Heights in a Short Train of Waves 



Series 
number 

n 


Wave number, m 


Total 
energy 

of 
group 


1 


2 


3 


4 


5 


6 


7 


1 
2 
3 
4 
5 
6 


1/2E 

3/4 

7/8 
15/16 
31/32 

63/64 

' 














1/2E 

2/2 

3/2 

4/2 

5/2 

6/2 


1/4E 

4/8 

11/16 

26/32 

57/64 


1/8E 
5/16 
16/32 
42/64 








1/16E 
6/32 
22/64 














1/32E 
7/64 

_ , ■ 


1/64E 







2-32 



With a large number of waves (a large n), energy decreases with 
increasing m, and the leading wave will eventually lose its identity. 
At the group center, energy increases and decreases rapidly - to nearly 
maximum and to nearly zero. Consequently, an energy front is located 
at the center wave group for deepwater conditions. If waves had been 
examined for shallow rather than deep water, the energy front would 
have been found at the leading edge of the group. For any depth, the 
ratio of group to phase velocity (C„/C) generally defines the energy 
front. Also, wave energy is transported in the direction of phase propa- 
gation, but moves with the group velocity rather than phase velocity. 

2,239 Summary - Linear Wave Theory . Equations describing water surface 
profile particle velocities, particle accelerations, and particle displace- 
ments for linear (Airy) theory are summarized in Figure 2-6. 

2.24 HIGHER ORDER WAVE THEORIES 

Solution of the hydroynamic equations for gravity-wave phenomena can 
be improved. Each extension of the theories usually produces better 
agreement between theoretical and observed wave behavior. The extended 
theories can explain phenomena such as mass transport that cannot be 
explained by linear theory. If amplitude and period are known precisely, 
the extended theories can provide more accurate estimates of such derived 
quantities as the velocity and pressure fields due to waves than can 
linear theory. In shallow water, the maximum wave height is determined 
by depth, and can be estimated without wave records. 

When concern is primarily with the oscillating character of waves, 
estimates of amplitude and period must be determined from empirical data. 
In such problems, the uncertainty about the accurate wave height and 
period leads to a greater uncertainty about the ultimate answer than does 
neglecting the effect of nonlinear processes. Thus it is unlikely that 
the extra work involved in using nonlinear theories is justified. 

The engineer must define regions where various wave theories are 
valid. Since investigators differ on the limiting conditions for the 
several theories, some overlap must be permitted in defining the regions. 
Le Mehaute (1969) presented Figure 2-7 to illustrate approximate limits 
of validity for several wave theories. Theories discussed here are indi- 
cated as are Stokes' third- and fourth-order theories. Dean (1973), 
after considering three analytic theories, presents a slightly different 
analysis. Dean (1973) and Le Mehaute (1969) agree in recommending cnoidal 
theory for shallow-water waves of low steepness, and Stokes' higher order 
theories for steep waves in deep water, but differ in regions assigned 
to Airy theory. Dean indicates that tabulated stream function theory is 
most internally consistent over most of the domain considered. For the 
limit of low steepness waves in transitional and deep water, the differ- 
ence between stream function theory and Airy theory is small. Additional 
wave theories not presented in Figure 2-7 may also be useful in studying 



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2-34 




^'^ 



0.001 



itiil:]ll[ii#i#ll''f^i'l-|!in']-]1^iti^|tiMr!^1-lin^i^^ 



I II 

0.01 0.02 0.04 0.06 0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 100 

d 



(ft/sec2) 



(after Le Mehaute,l969) 



Figure 2-7. Regions of Validity for Various Wove Theories 



2-35 



wave phenomena. For given values of H, d and T, Figure 2-7 may be used 
as a guide in selecting an appropriate theory. The magnitude of the 
Ursell parameter Ud shown in the figure may be used to establish the 
boiondaries of regions where a particular wave theory should be used. 
The Ursell parameter is defined by 

T 2 14 

"« = y^ . (2-45) 

For linear theory to predict accurately the wave characteristics, both 
wave steepness, H/gT^, and the Ursell parameter must be small as shown in 
Figure 2-7. 

2.25 STOKES' PROGRESSIVE, SECOND-ORDER WAVE THEORY 

Wave formulas presented in the preceding sections on linear wave 
theory are based on the assumption that the motions are so small that the 
free surface can be described to the first order of approximation by 
Equation 2-10: 

H /277X 27rt\ H 

v = — cos 1 — I = — cos or a cos d . 

2 \L T I 2 

More specifically, it is assumed that wave amplitude is small, and the 
contribution made to the solution by higher order terms is negligible. A 
more general expression would be: 

T? = acos(0) + a^Bj (L, d) cos (20) 

(2-46) 
+ a'Bj (L, d) cos(3fl) + • • • a"B„ (L, d) cos(n0) , 

where a = H/2, for first- and second-orders, but a < H/2 for orders 
higher than the second, and Bg, B3 etc. are specified functions of the 
wavelength L, and depth d. 

Linear theory considers only the first term on the right side of 
Equation 2-46. To consider additional terms represents a higher order of 
approximation of the free surface profile. The order of the approxima- 
tion is determined by the highest order term of the series considered. 
Thus, the ordinate of the free surface to the third order is defined by 
the first three terms in Equation 2-46. 

When the use of a higher order theory is warranted, wave tables, such 
as those prepared by Skjelbreia (1959), and Skjelbreia and Hendrickson 
(1962), should be used to reduce the possibility of numerical errors made 
in using the equations. Although Stokes (1880) first developed equations 
for finite amplitude waves, the equations presented here are those of 
Miche (1944). 



2-36 



2.251 Wave Celerity, Length, and Surface Profile . It can be shovm that, 
for second-order theories, expressions for wave celerity and wavelength 
are identical to those obtained by linear theory. Therefore, 



and 



^ gT , /27rd' 

C = 2- tanh 

2tt \ L 



L = ^^ tanh '-^ 
2n \ L 



(2-3) 



(2-4) 



The above equations, corrected to the third order, are given by: 



gT , /27rd^ 

C = — tanh 

271 \ L 



1 + 



and 



L = ^— tanh 

27r I L 



1 + 



T 



'nH\ 



5 + 2 cosh (47rd/L) + 2 cosh^ (47rd/L) 
8 sinh" (27rd/L) 



5 + 2 cosh (47rd/L) + 2 cosh^ (47rd/L) 



,(2-47) 



8 sinh" (2;rd/L) 



The equation of the free surface for second-order theory is 



(2-48) 



H /27rx 

— cos I 

2 \ L 



27rt ' 



+ 



(8L 



cosh (27rd/L) 
sinh^ (27rd/L) 



2 + cosh(47rd/L) 



cos 



u. 



t) 



For deep water, (d/L > 1/2) Equation 2-49 becomes, 

H„ J2-nx 2-ni 



■o I-..- --. ^ '^"o /4jrx 4fft\ 

T? = COS I — + cos — — - 

2 \L, TJ 4L^ \l^ t/ 



(2-49) 



(2-50) 



2.252 Water Particle Velocities and Displacements . The periodic x and z 
components of the water particle velocities to the second order are given 
by 



HgT cosh [27r(z + d)/L] /27rx lux^ 

U = ; ■ cos I — 

2L cosh(27rd/L) \ L T y 



3 /ttHV cosh [47r(z + d)/L] ^ttx 4jrtl 

+ — C . . ^ ,. TTTT cos 



(2-51) 



sinh* (27rd/L) 



2-37 



nU sinh [27r(z + d)/L] 2itx 

w = — C r~r-r: — 7r~^ — sin 1 

L sinh(27rd/L) \ L 



+ 1 i'^\ r si"h [4n(z + d)/L] . Mttx 
4 \l/ sinh'»(27rd/L) ''" \l 



(2-52) 



Second-order equations for water-particle displacements from their 
mean position for a finite amplitude wave are: 



? = - 



1 - 



HgT^ cosh [27r(z + d)/L] . Ilrrx _ 2-nt 
471 L cosh (27rd/L) ^"^ \ L T 



+ 



7:H^ 



8L sinh^ (2;rd/L) 



(2-53) 



3 cosh [47r(z + d)/L] 
2 sinh^ (27:d/L) 



Mtt: 



sin 



+ 



/ttHV Ct cosh [47r(z + d)/L] 
It) T sinhM27rd/L) 



and 



HgT^ sinh [27r(z + d)/L] 2nx l-nt 

t = —2 cos — 

^ 47rL cosh (27rd/L) \ L T 



3 ttH^ sinh [47r(z + d)/L] Mttx 

+ ~ — n. ; cos 

16 L sinh''(27rd/L) \L 



(2-54) 



2.253 Mass Transport Velocity . The last term in Equation 2-53 is of 
particular interest; it is not periodic, but is the product of time and 
a constant depending on the given wave period and depth. The term predicts 
a continuously increasing net particle displacement in the direction of 
wave propagation. The distance a particle is displaced during one wave 
period when divided by the wave period gives a mean drift velocity, U(z) , 
called the mass transport velocity. Thus, 



U(z) = 



/ttHV C cosh [47r(z + d)/L] 
T) 2 sinhM27rd/L) 



(2-55) 



Equation 2-53 indicates that there is a net transport of fluid by 
waves in the direction of wave propagation. If the mass transport, indi- 
cated by Equation 2-55 leads to an accumulation of mass in any region, the 
free surface must rise, thus generating a pressure gradient. A current, 
formed in response to this pressure gradient, will reestablish the distri- 
bution of mass. Studies of mass transport, theoretical and experimental, 
have been conducted by Longuet-Higgins (1953, 1960), Mitchim (1940), Miche 
(1944), Ursell (1953), and Russell and Osorio (1958). Their findings 
indicate that the vertical distribution of the mass transport velocity is 
modified so that the net transport of water across a vertical plane is 
zero. 



2-38 



2.254 Subsurface Pressure . The pressure at any distance below the fluid 
surface is given by 



H cosh [27r(z + d)/L] 
^§ 2 cosh (27rd/L) ^°' 




3 ttH^ tanh(27rd/L) 



8 



L sinhM27rd/L) 



cosh [47r(z + d)/L] 



sinh^ (27rd/L) 



cos r^ - ^ (2-56) 



ttH^ tanh (27rd/L) 



s'^T 



sinh^ (27rd/L) 



cosh 



4;r(z + d) 



2.255 Maximum Steepness of Progressive Waves . A progressive gravity wave 
is physically limited in height by depth and wavelength. The upper limit, 
or breaking wave height in deep water is a function of the wavelength, and 
in shallow and transitional water is a function of both depth and wavelength. 

Stokes (1880) predicted theoretically that a wave would remain stable 
only if the water particle velocity at the crest was less than the wave 
celerity or phase velocity. If the wave height were to become so large 
that the water particle velocity at the crest exceeded the wave celerity, 
the wave would become unstable and break. Stokes found that a wave having 
a crest angle less than 120 degrees would break (angle between two lines 
tangent to the surface profile at the wave crest). The possibility of 
existence of a wave having a crest angle equal to 120 degrees was shown by 
Wilton (1914) . Michell (1893) found that in deep water the theoretical 
limit for wave steepness was 



'H 



— 1 = 0.142 « - 
"olmax ' 



(2-57) 



Havelock (1918) confirmed Michell 's findings. 

Miche (1944) gives the limiting steepness for waves traveling in 
depths less than L^/2 without a change in form as 




27rd\ , /27rd\ 
= 0.142 tanh 



(2-58) 



Laboratory measurements by Danel (1952) indicate that the above 
equation is in close agreement with an envelope curve to laboratory 
observations. Additional discussion of breaking waves in deep and 
shoaling water is presented in Section 2.6, BREAKING WAVES. 

2.256 Comparison of the First- and Second-Order Theories . A comparison 
of first- and second-order theories is useful to obtain insight about the 



2-39 



choice of a theory for a particular problem. It should be kept in mind 
that linear (or first-order) theory applies to a wave which is symmetrical 
about Stillwater level and has water particles that move in closed orbits. 
On the other hand, Stokes' second-order theory predicts a wave form that 
is unsymmetrical about the Stillwater level but still symmetrical about a 
vertical line through the crest, and has water particle orbits which are 
open. 



************** EXAMPLE PROBLEM 



************** 



GIVEN : A wave traveling in water depth of d = 20 feet, with a wavelength 
of L = 200 feet and a height of H = 4 feet. 

FIND : 

(a) Compare the wave profiles given by the first- and second-order 
theories . 

(b) What is the difference between the first- and second- order 
horizontal velocities at the surface under both the crest and 
trough? 

(c) How far in the direction of wave propagation will a water particle 
move from its initial position during one wave period when z = 0? 

(d) What is pressure at the bottom under the wave crest as predicted 
by both the first- and second-order theories? 

(e) What is the wave energy per unit width of crest predicted by the 
first-order theory? 

SOLUTION : 

(a) The first-order profile Equation 2-10 is: 

H 

7? = — cos 6 , 
2 

where 

(2iTX 27rt 
L T 

and the second-order profile Equation 2-49 is: 



H ttH^ cosh(27rd/L) 

V = — cos 6 + — ; 

2 8L sinh^(27rd/L) 

for 

d 20 

- = = 0.1 , 

L 200 



2-40 



2 + cosh I 



cos 26 



and from Table C-2 



/27rd\ 
cosh t 1 = 1.2040 , 



sinh I 1 = 0.6705 , 



/47rd\ 
cosh = 1.8991 , 



L 

ttH^ cosh(2rrd/L) 



2 + cosh 



■ 47rd\ 



= 0.48 



8L sinh' (27rd/L) 

Therefore 

7? = 2 COS0 + 0.48 cos 2d , 

Tj, 2 = 2.48 ft., 

r?^ 2 = ~ 1-52 ft. 

where n^ 2 ^"'^ ^t 2 ^^^ ^^^ values of n at the crest 
(i.e. cos 9=1, cos 26 = 1) and trough (i.e. cos 6 = - 1, 
cos 26 = 1) according to second-order theory. 

Figure 2-8 shows the surface profile n as a function of 6. The 
second-order profile is more peaked at the crest and flatter at the 
trough than the first-order profile. The height of the crest above 
SWL is greater than one-half the wave height; consequently the 
distance below the SWL of the trough is less than one-half the 
height. Moreover, for linear theory, the elevation of the water 
surface above the SWL is equal to the elevation below the SWL; 
however, for second-order theory there is more height above SWL 
than below. 

(b) For convenience, let 

u^ 7 = value of u at crest according to first-order theory, 

^t 1 ~ v^l"® °^ " at a trough according to first-order theory, 

u„ 2 ~ value of u at a crest according to second-order theory, 

u+ 9 = value of u at a trough according to second-order theory. 



2-41 



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2-42 



According to first-order theory, a crest occurs at z = H/2, 

cos 6=1 and a trough at z = - H/2, cos = - 1. Equation 2-13 

therefore implies 



with 





= 


2 


T 
L 


cosh [27r(z + d)/L] 


"^c.l 


cosh (2;rd/L) 




H 










2 









and. 



with 



_ HgT cosh [27r(z + d)/L] 
^^'^ ~ 2L cosh(27rd/L) 



H 

z = — — 
2 



According to second-order theory, a crest occurs at z = n o = 
2.48 feet, cos = cos 29 = 1 and a trough at z = n^ 2 = "'l-52 
feet, cos e = - 1, cos 20 = 1. Equation 2-51 therefore implies 



u 



HgT cosh [27r(z + d)/L] 
'''■^ 2L cosh (27rd/L) 



+ 1 I'l^Y c cosh [47r(z + d)/L] 
4 \L / sinh" (27rd/L) 



with 



z = Vc2 = + 2.48 feet 



and. 



"f.2 = - 



HgT cosh [27r(z + d)/L] 
2L cosh (27rd/L) 



with 



4 \l / sinh''(27rd/L) 



z = ^^2 = ~~ 1-52 feet . 



2-43 



Entering Table C-2 with d/L = 0.10, find tanh (2TTd/L) = 0.5569. 

From Equation 2-3 which is true for both first- and second-order 
theories. 



C^ = — tanh 
27r 



27rd\ (32.2) (200) (0.5569) 



(t) = 



27r 



= 571 (ft/sec)' 



or 



C = 23.9 ft/sec. 



As a consequence. 



T 1 



= 0.0418 sec/ft 



L C 

Referring again to Table C-2, it is found that when 



H 
^=2 ' 



cosh 



27r(z + d) 



= cosh[27r(0.11)] = 1.249 



and when 



z = — — 



H 
2 ' 



cosh 



27r(z + d) 



= cosh [27r(0.09)] = 1.164 



Thus, the value of u at a crest and trough respectively according 
to first-order theory is 



\,1 = \ (32.2) (0.0418) ^^ = 2.8 ft/sec , 



4 1.164 

u, , = - (32.2) (0.0418) 

*'^ 2 1.2040 



= — 2.6 ft/sec 



2-44 



Entering Table C-2 again, it is found that when 



When 



z = 1?^ 2 = 2.48 feet 
27r(z + d) 



cosh 



cosh 



47r(z + d) 



= cosh [27r(0.1124)] = 1.260. 



= cosh [47r(0.1 124)] = 2.175 



z = T?^ 2 = ~ 1-52 feet , 



cosh 



cosh 



27r(z + d) 



47r(z + d) 



cosh [27r(0.0924)] = 1.174 



cosh [47r(0.0924)] = 1.753 



Thus, the value of u at a crest and trough respectively according 
to second-order theory is 

4 1.260 3 / 47r\^ 2.175 

u^2 = :; (32.2) (0.0418) — — - + - -— (23.9) -^^ = 3.6 ft/sec , 
''''' 2 1.2040 4 \200) 0.202 



4 1.174 

u,y = (32.2) (0.0418) — 

''2 1. 



3 / 47r\2 1.753 

+ -— (23 9) 

2040 4 \200/ ■ 0.202 



2.0 ft/ 



sec 



(c) To find the horizontal distance that a particle moves during one 
wave period at z = 0, Equation 2-55 can be written as: 



U(z) = 



AX(z) _ /ttH ^ C cosh [47r(z + d)/L] 
T ~ \ L 2 sinh^ (27rd/L) 



where AX(z) is the net horizontal distance traveled by a water 
particle, z feet below the surface, during one wave period. 



2-45 



For the example problem, when 
z = , 



AX(0) 



>= ^ (t)" 



cosh (47rd/L) C 
sinh2(27rd/L) ' 1. 



(ttH)^ cosh(47Td/L)_ (;r4)^ (1.899) 



2L sinh^(27rd/L) 2(200) (0.6705)' 



= 1.67 ft. 



(d) The first-order approximation for pressure under a wave is; 

pgH cosh [27r(z + d)/L] 



P = 



cosh (27rd/L) 



cos 6 — pgz . 



when 



and when 



6 = (i.e. the wave crest), cos = 1, 
27r(z + d) 



z = — d, cosh 



= cosh(O) =1.0 



Therefore 



(2) (32.2) (4) 1 
p = — — - (2) (32.2) (- 20) = 107 + 1288 = 1395 lbs/ ft^ 

M J. (^U^ 

at a depth of 20 feet below the SWL. The second-order terms 
according to Equation 2-56 are 



3 -nW tanh(27rd/L) 

8 ^^ IT sinh' (27rd/L) 



1 ttH^ tanh (27rd/L) 



cosh [47r(z + d)/L] 1 
sinh' (27rd/L) 3 



cos 26 



sinh' (27rd/L) 



cosh 



47r(z + d) 



Substituting in the equation: 

3 ,o^ .oo .. '^(4)' (0.5569) 

- (2) (32.2) 

8 200 (0.6705)' 



1 



(0.6705)' 



(1) 



-i (2) (32.2) ^_(i)- (0-5569) , , 
8 200 (0.6705)' 



2-46 



Thus, second-order theory predicts a pressure, 

p = 1395 + 14 = 1409 lbs/ft^ . 

(e) Using Equation 2-38, the energy in one wavelength per unit width 
of crest given by the first-order theory is: 

pgtfL (2) (32.2) (4)^200) ^^„^^ft-lbs 

E = ^ = ^^^-^^ i^jL_:^ .' = 25,800 —r- • 

8 8 ft 

Evaluation of the hydrostatic pressure component (1288 lbs/ft^) 
indicates that Airy theory gives a dynamic component of 107 lbs/ft^ 
while Stokes theory gives 121 lbs/ft^. Stokes theory shows a 
dynamic pressure component about 13 percent greater than Airy 
theory. 



2.26 CNOIDAL WAVES 



Long, finite-amplitude waves of permanent form propagating in shallow 
water are frequently best described by anoidal wave theory. The existence 
in shallow water of such long waves of permanent form may have first been 
recognized by Boussinesq , (1877) . However, the theory was originally 
developed by Korteweg and DeVries (1895) . The term anoidal is used since 
the wave profile is given by the Jacobian elliptical cosine function 
usually designated en . 

In recent years, cnoidal waves have been studied by many investigators. 
Wiegel (1960) summarized much of the existing work on cnoidal waves, and 
presented the principal results of Korteweg and DeVries (1895) and Keulegan 
and Patterson (1940) in a more usable form. Masch and Wiegel (1961) 
presented such wave characteristics as- length, celerity and period in 
tabular and graphical form, to facilitate application of cnoidal theory. 

The approximate range of validity for the cnoidal wave theory as 
determined by Laitone (1963) and others is d/L < 1/8, and the Ursell 
parameter, L^H/d^ > 26. (See Figure 2-7.) As wavelength becomes long, 
and approaches infinity, cnoidal wave theory reduces to the solitary wave 
theory which is described in the next section. Also, as the ratio of wave 
height to water depth becomes small (infinitesimal wave height), the wave 
profile approaches the sinusoidal profile predicted by the linear theory. 

Description of local particle velocities, local particle accelerations, 
wave energy, and wave power for cnoidal waves is difficult; hence their 
description is not included here, but can be obtained in graphical form 
from Wiegel (1960, 1964) and Masch (1964). 

Wave characteristics are described in parametric form in terms of the 
modulus k of the elliptic integrals. While k itself has no physical 



2-4- 



significance, it is used to express the relationships between the various 
wave parameters. Tabular presentations of the elliptic integrals and 
other important functions can be obtained from the above references. The 
ordinate of the water surface Xg measured above the bottom is given by 



Jt 



+ Hcn^ 



2K(k)^-| U 



(2-59a) 



where y^ is the distance from the bottom to the wave trough, en is the 
elliptic cosine function, K(k) is the complete elliptic integral of the 
first kind, and k is the modulus of the elliptic integrals. The argument 
of cn^ is frequently denoted simply by ( ), thus. Equation 2-59a above 
can be written as 



y^ + H cnM ) 



(2 -59b) 



The elliptic cosine is a periodic function where cn2[2K(k) ((x/L) - (t/T))] 
has a maximum amplitude equal to unity. The modulus k is defined over 
the range between and 1. When k = 0, the wave profile becomes a 
sinusoid as in the linear theory, and when k = 1, the wave profile 
becomes that of a solitary wave. 



The distance from the bottom to the wave trough. 
Equations 2-59a and b, is given by 



yt' 



as used in 



yj 
d 



d 



H 
d 



16d' , , H 

— — K(k) [K(k) - E(k)] + 1-7 
3V d 



(2-60) 



where 



is the distance from the bottom to the crest and E(k) is the 



complete elliptic integral of the second kind. Wavelength is given by 



L = 



'I6d^ 
3H 



kK(k) , 



(2-61) 



and wave period by 




kK(k) 



1 + 



H 



y? 



E(k)\ 
K(k) 



(2-62) 



Cnoidal waves are periodic and of permanent form thus L = CT. 

Pressure under a cnoidal wave at any elevation y, above the bottom 
depends on the local fluid velocity, and is therefore complex. However, 
it may be approximated in a hydrostatic form as 



p = Pg (y^ - y) 
2-48 



(2-63) 



that is, the pressure distribution may be assumed to vary linearly from 
pgyg at the bed to zero at the surface. 

Figures 2-9 and 2-10 show the dimensionless cnoidal wave surface 
profiles for various values of the square of the modulus of the elliptic 
integrals k^, while Figures 2-11 through 2-15 present dimensionless 
plots of the parameters which characterize cnoidal waves. The ordinates 
of Figures 2-11 and 2-12 should be read with care, since values of k^ 
are extremely close to 1.0 (k^ = l-lO"^ = 1-0. 1 = 0.99) . It is the 
exponent a of k^ = 1-10"" that varies along the vertical axis of 
Figures 2-11 and 2-12. 

Ideally, shoaling computations might best be performed using cnoidal 
wave theory since this theory best describes wave motion in relatively 
shallow (or shoaling) water. Simple, completely satisfactory procedures 
for applying cnoidal wave theory are not available. Although linear wave 
theory is often used, cnoidal theory may be applied by using figures such 
as 2-9 through 2-15. 

The following problem will illustrate the use of these figures. 

************** EXAMPLE PROBLEM ************** 

GIVEN : A wave traveling in water depth of d = 10 feet, with a period of 
T = 15 seconds, and a height of H = 2.5 feet. 

FIND : 

(a) Using cnoidal wave theory, find the wavelength L and compare this 
length with the length determined using Airy theory. 

(b) Determine the celerity C. Compare this celerity with the celerity 
determined using Airy theory. 

(c) Determine the distance above the bottom of the wave crest (y^) 
and wave trough (y^) . 

(d) Determine the wave profile. 
SOLUTION: 



(a) Calculate 



H 2.5 

T = — = 0.25 

d 10 



and 



'' ^ "' Jw - ^'■'' 



2-49 




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Figure 2-11. Relationship Between k*^, H /d and T vg/d 



2-52 





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Figure 2-14. Relationship Between T y g/d yf/d, H/yf and L^H/d^ 



2-55 




to 
■o 

X 

'^ 

■D 

c 
o 



c 

Qi 

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M6A 



2-56 



From Figure 2-11, entering with H/d and T/g/d', determine the square 
of the modulus of the complete elliptical integrals, k^. 

F = 1 - 10-^-9-y . 

Entering Figure 2-12 with the value of k^ gives 



or 








L = 


d^ 


190 , 






/I90d^" 


90(10)3 
2.5 










L = 


= 275.7 ft. 






From 


Airy 


the 


;ory 


* 









L=S^tanh(^'= 266.6 ft. 

To check whether the wave conditions are in the range for which 
cnoidal wave theory is valid, calculate d/L and the Ursell 
parameter, L^H/d^. 

^ ^° = 0.0363 < - O.K. , 



L 275.7 8 



L'H 1 /"i = 190 > 26 O.K. 



d^ idIL? \d 

Therefore, cnoidal theory is applicable, 
(b) Wave celerity is given by 

C = - = ^^ = 18.38 ft/sec. 
T 15 

while the Airy theory predicts 

L 266.6 

C = - = = 17.77 ft/sec. 

T 15 

Thus if it is assumed that the wave period is the same for Cnoidal 
and Airy theories then 

^Cnoidal _ ^Cnoidal ^ 
^Airy ^Airy 

2-57 



(c) The percentage of the wave height above the SWL may be determined 
from Figure 2-13. Entering the figure with L^H/d^ = 190, the 
value of (y^ - d)/H is found to be 0.833 or 83.3 percent. 
Therefore , 



Yc 



= 0.833 H + d. 



y^ = 0.833(2.5) + 10 = 2.083 + 10 = 12.083 feet . 
Also from Figure 2-13, 

(y, - d) 



H 

thus 



+ 1 = 0.833 



y^ 



= (0.833- 1) (2.5) + 10 = 9.583 ft. 



(d) The dimensionless wave profile is given on Figure 2-9 and is 
approximately the one drawn for k^ = i - 10"^. The results 
obtained in (c) above can also be checked by using Figure 2-9. 
For the wave profile obtained with k^ = 1 - 10"'*, it is seen that 
the SWL is approximately 0.17 H above the wave trough or 0.83 H 
below the wave crest. 

The results for the wave celerity determined under (b) above 
can now be checked with the aid of Figure 2-15. Calculate 

0.261 . 





H 2.5 




y, 9.583 


Enteri 


ng Figure 2-15 with 




^'" - 19( 


and 


d^ - ''" 




- = 0.261 



y^ 



it is found that 



C 

= 1.083 

gyf 



Therefore 



C = 1.083 V(32.2) (9.583y = 19 ft/sec 



2-58 



The difference between this number and the 18.38 ft/sec calculated 
under (b) above is the result of small errors in reading the 
curves . 



2.27 SOLITARY WAVE THEORY 

Waves considered in the previous sections were oscillatory or nearly 
oscillatory waves. The water particles move backward and forward with the 
passage of each wave, and a distinct wave crest and wave trough are evident, 
A solitary wave is neither oscillatory nor does it exhibit a trough. In 
the pure sense, the solitary wave form lies entirely above the Stillwater 
level. The solitary wave is a wave of translation relative to the water 
mass. 

Russell (1838, 1845) first recognized the existence of a solitary 
wave. The original theoretical developments were made by Boussinesq 
(1872), Lord Rayleigh (1876), and McCowan (1891), and more recently by 
Keulegan and Patterson (1940), Keulegan (1948), and Iwasa (1955). 

In nature it is difficult to form a truly solitary wave, because at 
the trailing edge of the wave there are usually small dispersive waves. 
However, long waves such as tsunamis and waves resulting from large dis- 
placements of water caused by such phenomena as landslides, and earthquakes 
sometimes behave approximately like solitary waves. When an oscillatory 
wave moves into shallow water, it may often be approximated by a solitary 
wave, (Munk, 1949). As an oscillatory wave moves into shoaling water, the 
wave amplitude becomes progressively higher; the crests become shorter and 
more pointed, and the trough becomes longer and flatter. 

The solitary wave is a limiting case of the cnoidal wave. When k^ = 
1, K(k) = K(l) = oo, and the elliptic cosine reduces to the hyperbolic 
secant function, y- = d, and Equation 2-59 reduces to 



d + H sech^ 



H 



(x - Ct) 



or 



1? = H sech^ 



3 H_ 

4 d^ 



(x - Ct) 



(2-64) 



where the origin of x is at the wave crest. The volume of water within 
the wave above the still water level per unit crest width is 



V = 



16 



% 



d' H 



(2-65) 



2-59 



An equal amount of water per unit crest length is transported 
forward past a vertical plane that is perpendicular to the direction of 
wave advance. Several relations have been presented to determine the 
celerity of a solitary wave; these equations differ depending on the 
degree of approximation. Laboratory measurements by Daily and Stephan 
(1953) indicate that the simple expression 

C = Vg (H + d)" , (2-66) 

gives a reasonably accurate approximation to the celerity. 

The water particle velocities for a solitary wave, as found by 
McCowan (1891) and given by Munk (1949), are 

1 + cos(My/d) cosh(Mx/d) ^^ ^^^ 

u = CN —'- (2-67) 

[cos(My/d) + cosh(Mx/d)]' ' 

sin(My/d) sinh (Mx/d) 

w = CN (2-68) 

[cos(My/d) + cosh(Mx/d)]^ 

where M and N are the functions of H/d shown on Figure 2-16, and y 
is measured from the bottom. The expression for horizontal velocity u, 
is often used to predict wave forces on marine structures sited in shallow 
water. The maximum velocity u^nox' occurs when x and t are both equal 
to zero; hence, 

CN 

"'"'« " 1 + cos(My/d) ■ ~^^' 

Total energy in a solitary wave is about evenly divided between 
kinetic and potential energy. Total wave energy per unit crest width is, 

E = ^^pgH3/2d3/2 , (2-70) 

and the pressure beneath a solitary wave depends upon the local fluid 
velocity as does the pressure under a cnoidal wave; however, it may be 
approximated by 

P = Pg(y,-y). (2-71) 

Equation 2-71 is identical to that used to approximate the pressure 
beneath a cnoidal wave. 

As a solitary wave moves into shoaling water it eventually becomes 
unstable and breaks. McCowan (1891) assumed that a solitary wave breaks 



2-60 




0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 

LI 

Relative wave height -r- 

(afterMunk,l949) 

Figure 2-16. Functions M and N in Solitary Wave Theory 



2-61 




when the water particle velocity at the wave crest becomes equal to the 
wave celerity. This occurs when 



(2-72) 



Laboratory investigations have shown that the value of (H/d)^^^^ = 0-78 
agrees better with observations for oscillatory waves than for solitary 
waves. Ippen and Kulin (1954) and Galvin (1969) have shown that the near- 
shore slope has a substantial effect on this ratio. Other factors such as 
bottom roughness may also be involved. For slopes of 0.0, 0.05, 0.10, and 
0.20, Galvin found that H/d ratios were approximately equal to 0.83, 
1.05, 1.19, and 1.32, respectively. Thus, it must be concluded that for 
some conditions. Equation 2-72 is unsatisfactory for predicting breaking 
depth. Further discussion of the breaking of waves with experimental 
results is in Section 2.6 - BREAKING WAVES. 

2.28 STREAM FUNCTION WAVE THEORY 

In recent years, numerical approximations to solutions of hydrodynamic 
equations describing wave motion have been proposed and developed by Dean 
(1965a, 1965b, 1967) and Monkmeyer (1970). The approach by Dean, termed 
a symmetric, stream function theory, is a nonlinear wave theory which is 
similar to higher order Stokes' theories. Both are constructed of suras of 
sine or cosine functions that satisfy the original differential equation 
(Laplace equation). The theory, however, determines the coefficient of 
each higher order term so that a best fit, in the least-squares sense, is 
obtained to the theoretically posed, dynamic, free-surface boundary con- 
dition. Assumptions made in the theory are identical to those made in the 
development of the higher-order Stokes' solutions. Consequently, some of 
the same limitations are inherent in the stream function theory; however, 
it represents a better solution to the equations used to approximate the 
wave phenomena. More important is that the stream function representation 
appears to better predict some of the wave phenomena observed in laboratory 
wave studies (Dean and L^Mehaut^, 1970), and may possibly describe naturally 
occurring wave phenomena better than other theories . 

The long tedious computations involved in evaluating the terms of the 
series expansions that make up the higher-order stream function solutions, 
make it desirable to use tabular or graphical presentations of the 
solutions. These tables, their use and range of validity have been 
developed by Dean (1973). 

2.3 WAVE REFRACTION 

2.31 INTRODUCTION 

Equation 2-2 shows that wave celerity depends on the depth of water 
in which the wave propagates. If the wave celerity decreases with depth, 
wavelength must decrease proportionally. Variation in wave velocity occurs 

2-62 



along the crest of a wave moving at an angle to underwater contours because 
that part of the wave in deeper water is moving faster than the part in 
shallower water. This variation causes the wave crest to bend toward align- 
ment with the contours, (See Figure 2-17.) This bending effect, called 
refraction, depends on the relation of water depth to wavelength. It is 
analogous to refraction for other types of waves such as, light and sound. 

In practice, refraction is important for several reasons such as: 

(1) Refraction, coupled with shoaling, determines the wave height 
in any particular water depth for a given set of incident deepwater wave 
conditions, that is wave height, period, and direction of propagation in 
deep water. Refraction therefore has significant influence on the wave 
height and distribution of wave energy along a coast. 

(2) The change of wave direction of different parts of the wave 
results in convergence or divergence of wave energy, and materially affects 
the forces exerted by waves on structures . 

(3) Refraction contributes to the alteration of bottom topography 
by its effects on the erosion and deposition of beach sediments. Munk and 
Traylor (1947) confirmed earlier work by many indicating the possible inter- 
relationships between refraction, wave energy distribution along a shore, 
and the erosion and deposition of beach materials. 

(4) A general description of the nearshore bathymetry of an area 
can sometimes be obtained by analyzing aerial photography of wave refraction 
patterns. While the techniques for performing such analyses are not well 
developed, an experienced observer can obtain a general picture of simple 
bottom topography. 

In addition to refraction caused by variations in bathymetry, waves 
may be refracted by currents, or any other phenomenon which causes one part 
of a wave to travel slower or faster than another part. At a coastal inlet, 
refraction may be caused by a gradient in the current. Refraction by a 
current occurs when waves intersect the current at an angle. The extent 
to which the current will refract incident waves depends on the initial 
angle between the wave crests and the direction of current flow, the 
characteristics of the incident waves, and the strength of the current. 
In at least two situations, wave refraction by currents may be of practical 
importance. At tidal entrances, ebb currents run counter to incident waves 
and consequently increase wave height and steepness. Also, major ocean 
currents such as the Gulf Stream may have some effect on the height, length 
and direction of approach of waves reaching the coasts. Quantitative 
evaluation of the effects of refraction by currents is difficult. Addi- 
tional research is needed in this area. No detailed discussion of this 
problem will be presented here, but an introduction is presented by 
Johnson (1947). 

The decrease in wave celerity with decreasing water depth can be con- 
sidered an analog to the decrease in the speed of light with an increase in 

2-63 




Figure 2-17. Wave Refraction at Westhampton Beach, Long Island, New York 



2-64 



the refractive index of the transmitting medium. Using this analogy, 
O'Brien (1942) suggested the use of Snell's law of geometrical optics for 
solving the problem of water-wave refraction by changes in depth. The 
validity of this approach has been verified experimentally by Chien (1954), 
Ralls (1956), and Wiegel and Arnold (1957). Chao (1970) showed analyti- 
cally that Format 's principle and hence Snell's law followed from the 
governing hydrodynamic equations, and was a valid approximation when applied 
to the refraction problem. Generally, two basic techniques of refraction 
analysis are available - graphical and numerical. Several graphical pro- 
cedures are available, but fundamentally all methods of refraction analyses 
are based on Snell's law. 

The assumptions usually made are: 

(1) Wave energy between wave rays or orthogonals remains constant. 
(Orthogonals are lines drawn perpendicular to the wave crests, and extend 

in the direction of wave advance.) (See Figure 2-17.) 

(2) Direction of wave advance is perpendicular to the wave crest, 
that is, in the direction of the orthogonals. 

(3) Speed of a wave of given period at a particular location 
depends only on the depth at that location. 

(4) Changes in bottom topography are gradual. 

(5) Waves are long-crested, constant-period, small -amplitude, and 
monochromatic. 

(6) Effects of currents, winds, and reflections from beaches, and 
underwater topographic variations, are considered negligible. 

2.32 GENERAL - REFRACTION BY BATHmETRY 

In water deeper than one-half the wavelength, the hyperbolic tangent 
function in the formula 

is nearly equal to unity, and Equation 2-2 reduces to 

" 2-n 

In this equation, the velocity C^, does not depend on depth; therefore 
in those regions deeper than one-half the wavelength (deep water) , refrac- 
tion by bathymetry will not be significant. Where the water depth is 
between 1/2 and 1/25 the wavelength (transitional water) , and in the region 
where the water depth is less than 1/25 the wavelength (shallow water), 
refraction effects may be significant. In transitional water, wave velocity 



2-65 



must be computed from Equation 2-2; in shallow water, tanh(2Trd/L) becomes 
nearly equal to 2TTd/L and Equation 2-2 reduces to Equation 2-9. 



^2 _ 



gd or C = (gd)*^ . (2-9) 



Both Equations 2-2 and 2-9 show the dependence of wave velocity on depth. 
To a first approximation, the total energy in a wave per unit crest width 
may be written as 



8 



(2-38) 



It has been noted that not all of the wave energy E is transmitted 
forward with the wave; only one-half is transmitted forward in deep water. 
The amount of energy transmitted forward for a given wave remains nearly 
constant as the wave moves from deep water to the breaker line if energy 
dissipation due to bottom friction (K^ = 1.0), percolation and reflected 
wave energy is negligible. 

In refraction analyses, it is assumed that for a wave advancing toward 
shore, no energy flows laterally along a wave crest; that is the transmitted 
energy remains constant between orthogonals. In deep water the wave energy 
transmitted forward across a plane between two adjacent orthogonals (the 
average energy flux) is 

^o = \KK^o' (2-73) 

where h^ is the distance between the selected orthogonals in deep water. 
The subscript o always refers to deepwater conditions. This power may 
be equated to the energy transmitted forward between the same two orthogonals 
in shallow water 



P = nbEC, (2-74) 



where b is the spacing between the orthogonals in the shallower water. 
Therefore, (1/2) b^ E^C^ = nb EC, or 

\/b„\/C„\ 

(2-75) 




From Equation 2-39, 







(2-76) 



2-66 



and combining Equations 2-75 and 2-76, 



H_ 
Ho 




(2-77) 



The term /(1/2) (1/n) (C^/C)' is known as the shoaling coefficient Kg 
or H/H^. This shoaling coefficient is a function of wavelength and water 
depth. Kg and various other functions of d/L, such as 2T7d/L, 47rd/L, 
tanh(27rd/L) , and sinh(4TTd/L) are tabulated in Appendix C, (Table C-1 for 
even increments of d/L^, and Table C-2 for even increments of d/L). 

Equation 2-77 enables determination of wave heights in transitional 
or shallow water, knowing the deepwater wave height when the relative 
spacing between orthogonals can be determined. The square root of this 
relative spacing, /b^/b', is the refraction coefficient K^. 

Various methods may be used for constructing refraction diagrams. 
The earliest approaches required the drawing of successive wave crests. 
Later approaches permitted the immediate construction of orthogonals, 
and also permitted moving from the shore to deep water (Johnson, O'Brien 
and Isaacs, 1948), (Arthur, et al . , 1952), (Kaplan, 1952) and (Saville 
and Kaplan, 1952) . 

The change of direction of an orthogonal as it passes over relatively 
simple hydrography may be approximated by 



sina, = M- 



sin a (Snell'slaw) (2-78) 



where ; 



a^ is the angle a wave crest (the perpendicular to an orthogonal) 
makes with the bottom contour over which the wave is passing, 

a^ is a similar angle measured as the wave crest (or orthogonal) 
passes over the next bottom contour, 

C, is the wave velocity (Equation 2-2) at the depth of the first 
contour, and 

C2 is the wave velocity at the depth of the second contour. 

From this equation, a template may be constructed which will show the 
angular change in a that occurs as an orthogonal passes over a 
particular contour interval, and construct changed-direction orthogonal. 
Such a template is shown in Figure 2-18. In application to wave refrac- 
tion problems, it is simplest to construct this template on a transparent 
material. 



2-67 



<J>a)^-^r)^f^vK>cJ — o 




o o oo o 







i/ 

C 

c. 
















*w| — 










r; 


^ 


E 


"h 


P- 


^ 


E 


3 


H 




4-> 

rH 

V 

H 

c 
o 

•H 
4J 
U 
0) 

m 



I 

CM 

u 



2-68 



Refraction may be treated analytically at a straight shoreline with 
parallel offshore contours, by using Snell's law directly: 

... -ig ... 

where a. is the angle between the wave crest and the shoreline, and ot^^ 
is the angle between the deepwater wave crest and the shoreline. 

For example, if cxq = 30° and the period and depth of the wave 
are such that C/C^ = 0.5, then 

a = sin-' [0.5 (0.5)] = 14.5 degrees 
cos a = 0.968 



and 



cosa^ = 0.866 




cosaA'^ / 0.866 \^ 



cos a / \ 0.968 



= 0.945 



Figure 2-19 shows the relationships between a, a^, period, depth, and 
K^ in graphical form. 

2.321 Procedures in Refraction Diagram Construction - Orthogonal Method . 
Charts showing the bottom topography of the study area are obtained. Two 
or more charts of differing scale may be required, but the procedures are 
identical for charts of any scale. Underwater contours are drawn on the 
chart, or on a tracing paper overlay, for various depth intervals. The 
depth intervals chosen depend on the degree of accuracy desired. If 
overlays are used, the shoreline should be traced for reference. In 
tracing contours, small irregularities must be smoothed out, since bottom 
features that are comparatively small in respect to the wavelength do not 
affect the wave appreciably. 

The range of wave periods and wave directions to be investigated is 
determined by a hindcasting study of historical weather charts or from 
other historical records relating to wave period and direction. For each 
wave period and direction selected, a separate diagram must be prepared. 
C1/C2 values for each contour interval may then be marked between contours, 
The method of computing C1/C2 is illustrated by Table 2-2; a tabulation 
of C1/C2 for various contour intervals and wave periods is given in 
Table C-4 of Appendix C. 

To construct orthogonals from deep to shallow water, the deepwater 
direction of wave approach is first determined. A deepwater wave front 
(crest) is drawn as a straight line perpendicular to this wave direction. 



2-69 




(O 


o 




o 






o 










j: 


TT 




*- 


O 




S 


O 


to 








a> 






a. 




o 


o 




J 


1/5 












c 


O 




o 


o 




c 






o 

o 

o 


o 
d 




a: 
o 

w 


<o 




3 


o 




o 


o 






o 






'j- 






o 






o 




X = 


o 


rj 


o 




T5 ^- 


c c 




o» 


°- 


CVJ 




o ^ 






O 
O 






o 




Q — 

> — 
o o 


o 




5 n 


o 
d 






(D 




<1> -c 


O 




CJ> o> 


O 




= '^ 


O 




o 2 


o 




f ^ 






OWJ 


o 






o 




CD 


o 






o 




a> 


o 




3 


o 




Ll. 


o 






o 






o 






o 






o 







o 



o 



2-70 



and suitably spaced orthogonals are drawn perpendicular to this wave front 
and parallel to the chosen direction of wave approach. Closely spaced 
orthogonals give more detailed results than widely spaced orthogonals. 
These lines are extended to the first depth contour shallower than L^/2 
where L^ (in feet) = 5.12 T^. 

TABLE 2-2 EXAMPLE COMPUTATIONS OF VALUES OF 
Cj/C2 FOR REFRACTION ANALYSIS 



T = 10 seconds 


(1) (2) 


(3) 


(4) 


(5) 


d 
(ft) 


d 


2ud 

tanh 

L 


Ci 
C2 


C2 


6 

12 
18 
24 


0.0117 
0.0234 
0.0352 
0.0469 


0.268 

0.374 
0.453 
0.516 


1.40 
1.21 
1.14 


0.72 
0.83 
0.88 



Column 1 gives depths corresponding to chart contours . These 
would extend from 6 feet to a depth equal to L /2 . 

Column 2 is column 1 divided by L^ corresponding to the given 
period. 

Column 3 is the value of tanh 27rd/L found in Table C-1 of 
Appendix C, corresponding to the value of d/L^ in 
column 2. This term is also C/C^^. 

Column 4 is the quotient of successive terms in column 3. 

Column 5 is the reciprocal of column 4. 

2.322 Procedure when a is Less than 80 Degrees . Recall that a is the 

angle a wave crest makes with the bottom contour. Starting with any one 

orthogonal and using the refraction template in Figure 2-18, the following 
steps are performed in extending the orthogonal to shore: 

(a) Sketch a contour midway between the first two contours to be 
crossed, extend the orthogonal to the midcontour, and construct a tangent 
to the midcontour at this point. 

(b) Lay the line on the template labelled orthogonal along the in- 
coming orthogonal with the point marked 1.0 at the intersection of the 
orthogonal and midcontour (Figure 2-20 top) ; 

2-71 



Contour 



Templote Orthogonal Line 



"-X 



20'"^ 



Tangent to Mid- ^ 'Ci 



Contour 






bO 



\^ 



Incoming Orttiogonol 




'"'^Wc.-r-l- /~ 



Turning Point- 



Contour 
20 ^^ 



Template Orttiogonal' Line 



Turned Orthogonal 



II- 



1071 



Tangent to Mid- 
Contour -^ 



^o 



\<^ 



1.0461 




Incoming Orthogonal 



NOTE: The template has been turned obout R until the volue '/r ; 1.045 

intersects the tangent to the mid-contour. The templote "orthogonal line 
lies in the direction of the turned orthogonol. This direction is to be laid 
off ot some point " B"on the incoming orthogonol which is equidistant 
from the two contours along the incoming and outgoing orthogonals. 

Figure 2-20. Use of the Refraction Template 



2-72 



(c) Rotate the template about the turning point until the C2/C2 value 
corresponding to the contour interval being crossed intersects the tangent 
to the midcontour. The orthogonal line on the chart now lies in the direc- 
tion of the turned orthogonal on the template (Figure 2-20 bottom) ; 

(d) Place a triangle along the base of the template and construct 
a perpendicular to it so that the intersection of the perpendicular with 
the incoming orthogonal is midway between the two contours when the dis- 
tances are measured along the incoming orthogonal and the perpendicular 
(See Point B in Figure 2-20 bottom) . Note that this point is not neces- 
sarily on the midcontour line. This line represents the turned orthogonal; 

(e) Repeat the above steps for successive contour intervals . 

If the orthogonal is being constructed from shallow to deep water, the 
same procedure may be used, except that C2/C]^ values are used instead of 

A template suitable for attachment to a drafting machine can be made. 
Palmer (1957) , and may make the procedure simpler if many diagrams are to 
be used. 

2.323 Procedure when a is Greater than 80 Degrees - The R/J Method . In 
any depth, when a becomes greater than 80 degrees, the above procedure 
cannot be used. The orthogonal no longer appears to cross the contours, 
but tends to run almost parallel to them. In this case, the contour 
interval must be crossed in a series of steps. The entire interval is 
divided into a series of smaller intervals. At the midpoint of the indi- 
vidual subintervals, orthogonal-angle turnings are made. 

Referring to Figure 2-21, the interval to be crossed is divided into 
segments or boxes by transverse lines. The spacing R, of the transverse 
lines is arbitrarily set as a ratio of the distance J, between the con- 
tours. For the complete interval to be crossed, C2/C1 is computed or 
found from Table C-4 of Appendix C. (C2/C^, not 0.-^/^2-^ 

On the template (Figure 2-18), a graph showing orthogonal angle 
turnings Aa, is plotted as a function of the C2/C1 value for various 
values of the ratio R/J. The Aa value is the angle turned by the in- 
coming orthogonal in the center of the subinterval . 

The orthogonal is extended to the middle of the box, Aa is read 
from the graph, and the orthogonal turned by that angle. The procedure 
is repeated for each box in sequence, until a at a plotted or interpo- 
lated contour becomes smaller than 80 degrees. At this point, this method 
of orthogonal construction must be stopped, and the preceding technique 
for a smaller than 80 degrees used, otherwise errors will result. 

2.324 Refraction Fan Diagrams . It is often convenient, especially where 
sheltering land forms shield a stretch of shore from waves approaching in 
certain directions, to construct refraction diagrams from shallow water 



2-73 



J = Distance between contours at turning points,* 

R = Distance along orttfogonal 

T = 12 seconds 

Lo= 737 ft. 

For contour interval from 40fm to 30 fm C /C = 1.045, 0/0=0.957 

12 2 1 




Acc = 20-25 



Figure 2-21. Refraction Diagram Using R/J Method 



2-74 



toward deep water. In such cases, a sheaf or fan of orthogonals may be 
projected seaward in directions some 5 or 10 degrees apart. See Figure 
2-22a. With the deepwater directions thus determined by the individual 
orthogonals, companion orthogonals may be projected shoreward on either 
side of the seaward projected ones to determine the refraction coefficient 
for the various directions of wave approach. (See Figure 2 -22b.) 

2.325 Other Graphical Methods of Refraction Analysis . Another graphical 
method for the construction of refraction diagrams is the wave-front 
method (Johnson, et al., 1948). This method is particularly applicable 
to very long waves where the crest alignment is also desired. The method 
is not presented here, where many diagrams are required, because, where 
many diagrams are required, it is overbalanced by the advantages of the 
orthogonal method. The orthogonal method permits the direct construction 
of orthogonals and determination of the refraction coefficient without the 
intermediate step of first constructing successive wave crests. Thus, 
when the wave crests are not required, significant time is saved by using 
the orthogonal method. 

2.326 Computer Methods for Refraction Analysis . Harrison and Wilson (1964) 
developed a method for the numerical calculation of wave refraction by use 
of an electronic computer. Wilson (1966) extended the method so that, in 
addition to the numerical calculation, the actual plotting of refraction 
diagrams is accomplished automatically by use of a computer. Numerical 
methods are a practical means of developing wave refraction diagrams when 

an extensive refraction study of an area is required, and when they can 
be relied upon to give accurate results. However, the interpretation of 
computer output requires care, and the limitations of the particular scheme 
used should be considered in the evaluation of the results. For a dis- 
cussion of some of these limitations, see Coudert and Raichlen (1970). 
For additional references, the reader is referred to the works of Keller 
(1958), Mehr (1962), Griswold (1963), Wilson (1966), Lewis, et al., (1967), 
Dobson (1967), Hardy (1968), Chao (1970), and Keulegan and Harrison (1970), 
in which a number of available computer programs for calculation of refrac- 
tion diagrams are presented. Most of these programs are based on an algo- 
rithm derived by Munk and Arthur (1951) and, as such, are fundamentally 
based on the geometrical optics approximation. (Fermat's Principle.) 

2.327 Interpretation of Results and Diagram Limitations . Some general 
observations of refraction phenomena are illustrated in Figures 2-23, 24, 
and 25. These figures show the effects of several common bottom features 
on passing waves. Figure 2-23 shows the effect of a straight beach with 
parallel evenly spaced bottom contours on waves approaching from an angle. 
Wave crests turn toward alignment with the bottom contours as the waves 
approach shore. The refraction effects on waves normally incident on a 
beach fronted by a submarine ridge or submarine depression are illustrated 
in Figure 2-24a and 2-24b. The ridge tends to focus wave action toward 
the section of beach where the ridge line meets the s horeli ne. The ortho- 
gonals in this region are more closely spaced; hence-v/b^/o is greater 
than 1.0 and the waves are higher than they would be if no refraction 
occurred. Conversely, a submarine depression will cause orthogonals to 



2-75 




u 



C 

o 

■H 
■P 

o 

ni 
^( 

a> 
ai 

a> 
P- 
X 

I 

C 

uu 

O 

O 
t/) 

3 



(N 

I 

(Nl 

u 



2-76 



Beach Line 



'Breakers 



»*JL-'X«'L^xxx,xxxr;x)i»^*»* _x1 




Figure 2-23. Refraction Along a Straight Beach with Parallel 
Bottom Contours 



Beach Line 




Contours 




Orthogonals 



Beach Line 




Contours 



Orthogonals 



(a) (b) 

Figure 2-24. Refraction by a Submarine Ridge (a) and Submarine 
Canyon (b) 




Figure 2-25. Refraction Along an Irregular Shoreline 



2-77 



diverge, resulting in low heights at the shore. (b^i/b less than 1.0.) 
Similarly, heights will be greater at a headland than in a bay. Since 
the wave energy contained between two orthogonals is constant, a larger 
part of the total energy expended at the shore is focused on projections 
from the shoreline; consequently, refraction straightens an irregular 
coast. Bottom topography can be inferred from refraction patterns on 
aerial photography. The pattern in Figure 2-17 indicates the presence 
of a submarine ridge. 

Refraction diagrams can provide a measure of changes in waves 
approaching a shore. However, the accuracy of refraction diagrams is 
limited by the validity of the theory of construction and the accuracy 
of depth data. The orthogonal direction change (Equation 2-78) is 
derived for straight parallel contours. It is difficult to carry an 
orthogonal accurately into shore over complex bottom features (Munk and 
Arthur, 1951). Moreover, the equation is derived for small waves moving 
over mild slopes. 

Dean (1973) considers the combined effects of refraction and shoal- 
ing including nonlinearities applied to a slope with depth contours 
parallel to the beach but not necessarily of constant slope. He finds 
that non-linear effects can significantly increase (in coirparison with 
linear theory) both amplification and angular turning of waves of low 
steepness in deep water. 

Strict accuracy for height changes cannot be expected for slopes 

steeper than 1:10, although model tests have shown that direction 

changes nearly as predicted even over a vertical discontinuity (Wiegel 
and Arnold, 1957). Accuracy where orthogonals bend sharply or exhibit 
extreme divergence or convergence is questionable because of energy 
transfer along the crest. The phenomenon has been studied by Beitinjani 
and Brater (1965), Battjes (1968) and Whalin (1971). Where two ortho- 
gonals meet, a caustic develops. A caustic is an envelope of ortho- 
gonal crossings, caused by convergence of wave energy at the caustic 
point. An analysis of wave behavior near a caustic is not available; 
however, qualitative analytical results show that wave amplitude decays 
exponentially away from a caustic in the shadow zone, and there is a 
phase shift of it/2 across the caustic (Whalin 1971). Wave behavior 
near a caustic has also been studied by Pierson (1950), Chao (1970) and 
others. Little quantitative information is available for the area 
beyond a caustic. 

2.328 Refraction of Ocean Waves . Unlike Monochromatic waves, actual 
ocean waves are more complicated. Their crest lengths are short; their 
form does not remain permanent; and their speed, period, and direction 
of propagation vary from wave to wave. 

2-78 



Pierson (1951), Longuet-Higgins (1957), and Kinsman (1965), have sug- 
gested a solution to the ocean-wave refraction problem. The sea surface 
waves in deep water become a number of component monochromatic waves, each 
with a distinct frequency and direction of propagation. The energy spec- 
trum for each component may then be found and the conventional refraction 
analysis techniques applied. Near the shore, the wave energy propagated 
in a particular direction is approximated as the linear sum of the spectra 
of wave components of all frequencies refracted in the given direction from 
all of the deepwater directional components. 

The work required for this analysis, even for a small number of indi- 
vidual components, is laborious and time consuming. More recent research 
by Borgman (1969) and Fan and Borgman (1970), has used the idea of direc- 
tional spectra which may provide a technique for solving complex refraction 
problems more rapidly. 

2.4 WAVE DIFFRACTION 

2.41 INTRODUCTION 

Diffraction of water waves is a phenomenon in which energy is trans- 
ferred laterally along a wave crest. It is most noticeable where an other- 
wise regular train of waves is interrupted by a barrier such as a breakwater 
or an islet. If the lateral transfer of wave energy along a wave crest and 
across orthogonals did not occur, straight, long-crested waves passing the 
tip of a structure would leave a region of perfect calm in the lee of the 
barrier, while beyond the edge of the structure the waves would pass un- 
changed in form and height. The line separating two regions would be a 
discontinuity. A portion of the area in front of the barrier would, how- 
ever, be disturbed by both the incident waves and by those waves reflected 
by the barrier. The three regions are shown in Figure 2-26a for the hypo- 
thetical case if diffraction did not occur, and in Figure 2-26b for the 
actual phenomenon as observed. The direction of the lateral energy trans- 
fer is also shown in Figure 2-26a. Energy flow across the discontinuity 
is from Region II into Region I. In Region III, the superposition of 
incident and reflected waves results in the appearance- of short-crested 
waves if the incident waves approach the breakwater obliquely. A partial 
standing wave will occur in Region III if the waves approach perpendicular 
to the breakwater. 

This process is also similar to that for other types of waves, such 
as light or sound waves. 

Calculation of diffraction effects is important for several reasons. 
Wave height distribution in a harbor or sheltered bay is determined to 
some degree by the diffraction characteristics of both the natural and 
manmade structures affording protection from incident waves. Therefore, 
a knowledge of the diffraction process is essential in planning such 
facilities. Proper design and location of harbor entrances to reduce 
such problems as silting and harbor resonance also require a knowledge 
of the effects of wave diffraction. The prediction of wave heights near 



2-79 




Figure 2-26a. Wave Incident on a Breakwater 
No Diffraction 




Figure 2- 26b. Wave Incident on a Breakwater 
Diffraction Effects 



2-80 



the shore is affected by diffraction caused by naturally occurring changes 
in hydrography. An aerial photograph illustrating the diffraction of 
waves by a breakwater is shown in Figure 2-27. 

Putnam and Arthur (1948) presented experimental data verifying a 
method of solution proposed by Penny and Price (1944) for wave behavior 
after passing a single breakwater. Wiegel (1962) used a theoretical 
approach to study wave diffraction around a single breakwater. Blue and 
Johnson (1949) dealt with the problem of the behavior of waves after 
passing through a gap, as between two breakwater arms. 

The assumptions usually made in the development of diffraction 
theories are: 

(1) Water is an ideal fluid, i.e., inviscid and incompressible. 

(2) Waves are of small -amplitude and can be described by linear 
wave theory. 

(3) Flow is irrotational and conforms to a potential function which 
satisfies the Laplace equation. 

(4) Depth shoreward of the breakwater is constant. 

2.42 DIFFRACTION CALCULATIONS 

2.421 Waves Passing a Single Breakwater . From a presentation by Wiegel 
(1962), diffraction diagrams have been prepared which, for a uniform depth 
adjacent to an impervious structure, show lines of equal wave height re- 
duction. These diagrams are shown in Figures 2-28 through 2-39; the graph 
coordinates are in units of wavelength. Wave height reduction is given in 
terms of a diffraction coefficient K' which is defined as the ratio of a 
wave height H, in the area affected by diffraction to the incident wave 
height H^, in the area unaffected by diffraction. Thus, H and H^ are 
determined by H = K'H^. 

The diffraction diagrams shown in Figures 2-28 through 2-39 are con- 
structed in polar coordinate form with arcs and rays centered at the struc- 
ture's tip. The arcs are spaced one radius -wavelength unit apart and rays 
15 apart . In application, a given diagram must be scaled up or down so 
that the particular wavelength corresponds to the scale of the hydrographic 
chart being used. Rays and arcs on the refraction diagrams provide a 
coordinate system that makes it relatively easy to transfer lines of 
constant K' on the scaled diagrams. 

When applying the diffraction diagrams to actual problems, the wave- 
length must first be determined based on the water depth at the tip of the 
structure. The wavelength L, in water depth dg, may be found by com- 
puting dg/L^ = dg/5.12T2 and using Appendix C, Table C-1 to find the 
corresponding value of dg/L. Dividing dg by dg/L will give the shallow 
water wave length L. It is then useful to construct a scaled diffraction 

2-81 




Figure 2-21 . Wave Diffraction at Channel Islands Harbor Breakwater, 
California 



2-82 



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2-94 



diagram overlay template to correspond to the hydrographic chart being 
used. In constructing this overlay, first determine how long each of its 
radius-wavelength units must be. As noted previously, one radius -wave length 
unit on the overlay must be identical to one wavelength on the hydrographic 
chart. The next step is to construct and sketch all overlay rays and arcs 
on clear plastic or translucent paper. This allows penciling in of the 
scaled lines of equal K for each angle of wave approach that may be 
considered pertinent to the problem. Thus, after studying the wave field 
for one angle of wave approach, K lines may be erased for a subsequent 
analysis of a different angle of wave approach. 

The diffraction diagrams in Figures 2-28 through 2-39 show the break- 
water extending to the right as seen looking toward the area of wave dif- 
fraction; however, for some problems the structure may extend to the left. 
All diffraction diagrams presented may be reversed by simply turning the 
transparency over to the opposite side. 

Figure 2-40 illustrates the use of a template overlay. Also indicated 
is the angle of wave approach which is measured counterclockwise from the 
breakwater. This angle would be measured clockwise from the breakwater if 
the diagram were turned over. Figure 2-40 also shows a rectangular coordi- 
nate system with distance expressed in units of wavelength. Positive 
x direction is measured from the structure's tip along the breakwater and 
positive y direction is measured into the diffracted area. 




Template Overlay 



Angle of Wave Approach 



Breakwater 



f'/'/'/'r /=z= 



I. O.I. I. '././. 1. 1. 1. '.I. '.I. J. '7^ 



-•-X 



Wave Crests 



Figure 2-40. Diffraction for a Single Breakwater Normal Incidence 

The following problem illustrates determination of a single wave 
height in the diffraction area. 
************** EXAMPLE PROBLEM ************** 

GIVEN : Waves with a period of T = 8 seconds and height of H = 10 feet 
impinge upon a breakwater at an angle of 135 degrees. The water depth 
at the tip of the breakwater toe is dg = 15 feet. Assume that one inch 
on the hydrographic chart being used is equivalent to 133 feet. 



2-95 



FIND: The wave height at a point P having coordinates in units of wave- 
length of X = 3 and y = 4. (Polar coordinates of x and y are r=5 at 53°.) 

SOLUTION : 

Since dg = 15 feet, T = 8 seconds, 



Using Table C-1 with 



d. 


^' ^^ -0 0153 


Lo 


5.12T^ (5.12) (64) 


th 






d d. 




— = — = 0.0458 
o o 



the corresponding value of 



d d, 

- or — = 0.0899 , 
L L 



therefore. 



d. 



(V. 



= 167 feet 



0.0899 



Because 1 inch represents 133 feet on the hydrographic chart and 

L = 167 feet, the wave-length is 1.26 inches on the chart. 

This provides the necessary information for scaling Figure 2-36 to the 

hydrographic chart being used. Thus 1.26 inches represents a radius/ 

wavelength unit. 

For this example, point P and those lines of equal K' situated 
nearest P are shown on a schematic overlay. Figure 2-41. This over- 
lay is based on Figure 2-36 since the angle of wave approach is 135 . 
It should be noted that Figure 2-41, being a schematic rather than a 
true representation of the overlay, is not drawn to the hydrographic 
chart scale calculated in the problem. From Figure 2-41 it is seen 
that K' at point P is approximately 0.086. Thus the diffracted 
wave height at this point is 

H = K'H,. = (0.086) (10) = 0.86 foot say 0.9 foot . 

The above calculation indicates that a wave undergoes a substantial 
height reduction in the area considered. 

************************************* 



2-96 



OVERLAY 
(Figure 2-36) 



X and y are measured in units 
of wavelength. 
(These units vory depending 
on the wavelength and the 
chart scale.) 



180' 




Wave Crests 



Direction of Wave Approach 



Figure 2-41. Schematic Representation of Wave Diffraction Overlay 



2-97 



2.422 Waves Passing a Gap of Width Less than Five Wavelengths at Normal 
Incidence . The solution of this problem is more complex than that for a 
single breakwater, and it is not possible to construct a single diagram 
for all conditions. A separate diagram must be drawn for each ratio of 
gap width to wavelength B/L. The diagram for a B/L-ratio of 2 is shown 
in Figure 2-42 which also illustrates its use. Figures 2-43 through 2-52 
(Johnson, 1953) show lines of equal diffraction coefficient for B/L-ratios 
of 0.50, 1.00, 1.41, 1.64, 1.78, 2.00, 2.50, 2.95, 3.82 and 5.00. A 
sufficient number of diagrams have been included to represent most gap 
widths encountered in practice. In all but Figure 2-48 (B/L = 2.00), the 
wave icrest lines have been omitted. Wave crest lines are usually of use 
only for illustrative purposes. They are, however, required for an 
accurate estimate of the combined effects of refraction and diffraction. 
In such cases, wave crests may be approximated with sufficient accuracy 
by circular arcs. For a single breakwater, the arcs will be centered on 
the breakwater tip. That part of the wave crest extending into unprotected 
water beyond the K' =0.5 line may be approximated by a straight line. 

For a breakwater gap, crests that are more than eight wavelengths behind 
the breakwater may be approximated by an arc centered at the middle of 
the gap; crests to about six wavelengths may be approximated by two arcs, 
centered on the two ends of the breakwater and may be connected by a 
smooth curve (approximated by a circular arc centered at the middle of 
the gap) . Only one-half of the diffraction diagram is presented on the 
figures since the diagrams are symmetrical about the line x/L = 0. 

2.423 Waves Passing a Gap of Width Greater Than Five Wavelengths at 
Normal Incidence . Where the breakwater gap width is greater than five 
wavelengths, the diffraction effects of each wing are nearly independent, 
and the diagram (Figure 2-33) for a single breakwater with a 90° wave 
approach angle may be used to define the diffraction characteristic in 
the lee of both wings (See Figure 2-53.) 

2.424 Diffraction at a Gap-Oblique Incidence . When waves approach at an 
angle to the axis of a breakwater, the diffracted wave characteristics 
differ from those resulting when waves approach normal to the axis. An 
approximate determination of diffracted wave characteristics may be 
obtained by considering the gap to be as wide as its projection in the 
direction of incident wave travel as shown in Figure 2-54. Calculated 
diffraction diagrams for wave approach angles of 0°, 15°, 30°, 45°, 60° 
and 75° are shown in Figures 2-55, 56 and 57. Use of these diagrams will 
give more accurate results than the approximate method. A comparison of 

a 45° incident wave using the approximate method and the more exact diagram 
method is shown in Figure 2-58. 

2.43 REFRACTION AND DIFFRACTION COMBINED 

Usually the bottom seaward and shoreward of a breakwater is not 
flat; therefore, refraction occurs in addition to diffraction. Although 
a general unified theory of the two has not yet been developed, some in- 
sight into the problem is presented by Battjes (1968) . An approximate 
picture of wave changes may be obtained by: (a) constructing a refraction 

2-98 




u 
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o 

B OQ 
nj ^-^ 

OO c/l 

•r-l 4-> 
Q 00 

c 



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4-1 O 



4-1 

T3 O 

N ^ 

•H 4-> 

(L> 

(U nJ 

L3 O 



I 

u 

3 



2-99 



y/L 

..0 2 4- 6 8 10 (2 14 16 18 20 


















K'=0.l 




^^ 


6 




-^ 






































5i 

4 










































Dl SECTION OF , 
INirOENT W4VE 

k2 


























\ 
















if 


^^r\ 


^ 


N 


^ 






GEOMETRIC 


SHADOW 






r 

GAP — 1 
8=1-/2 p 


■50\ \ 


\ 




1 















10 

o 




M 

d 










(Johnson, 1952) 



Figure 2-43. Contours of Equal Diffraction Coefficient 
Gap Width =0.5 Wave Length (B/L =0.5) 



.^ Diffrocfed Wove Height 
Incident Wove Height 



S cj d d o 

M H n N ■* 

at V V V V 




(Johnson, 1952) 

Figure 2-44. Contours of Equal Diffraction Coefficient 
Gap Width = 1 Wave Length (B/L = 1) 



2-100 




GAP 

B'UIL K =1.145 K:|.0 K'=0.8 K'=0.6 
" DIRECTION OF 



INCIDENT WAVE 



K'eO.4 



( Johnson, 1952 ) 



Figure 2-45. Contours of Equal Diffraction Coefficient 
Gap Width =1.41 Wave Lengths (B/L = 1.41) 



DIRECTION OF 
INCIDENT WAVE 



, Diffrocted Wove Height 
Incident Wove Height 



K'«0.4 




( Johnson, 1952) 

Figure 2-46. Contours of Equal Diffraction Coefficient 
Gap Width =1.64 Wave Lengths (B/L = 1.64) 



2-101 




PI DIRECTION OF , 
', INCIDENT WOVF 



( Johnson, 1952) 



Figure 2-47. Contours of Equal Diffraction Coefficient 
Gap Width =1.78 Wave Lengths (B/L = 1.78) 



,_ Diffrocted Wove Height 
Incident Wave Height 




(Johnson, 1952) 



Figure 2-48. 



Contours of Equal Diffraction Coefficient 
Gap Width = 2 Wave Lengths (B/L = 2) 



2-102 




GAP 
B=2.50L 



I K'=l.2 

I DIRECTION OF , 

* ra INCIDENT WAVE 



K'= 1.0 



K'=0.8 



K =0.6 

( Johnson, 1952) 



Figure 2-49. Contours of Equal Diffraction Coefficient 
Gap Width =2.50 Wave Lengths (B/L = 2.50) 



DIRECTION OF 
INCIDENT WAV^ 



,_ Diffrocted Wove Height 
Incident Wave Height 




( Johnson, 1952) 

Figure 2-50. Contours of Equal Diffraction Coefficient 
Gap Width =2.95 Wave Lengths (B/L = 2.95) 



2-103 




ri 



(Johnson, 1952) 



'BREAKWATER 



Figure 2-51. Contours of Equal Diffraction Coefficient 
Gap Width =3.82 Wave Lengths (B/L = 3.82) 



i^ 



, Diffrocted Wove Height 
'^ ■ Incident Wove Height 



GAP 
B:SL 



DIRECTION OF 
INCIDENT WAVE ^ 

K'sl.O K'sl.O 



K'« 1.282 



K't|.2 




( Johnson, 1952) 

Figure 2-52. Contours of Equal Diffraction Coefficient 
Gap Width = 5 Wave Lengths (B/L = 5) 



2-104 



Template Overlays 




Wave Crests 



Figure 2-53. Diffraction for a Breakwater Gap of Width > 5L (B/L > 5) 




YZZZZZZZZZZZZZ. 
Breakwater 



maginary Equivalent Gap 



Wave Crests 



( Johnson, 1952) 
Figure 2-54. Wave Incidence Oblique to Breakwater Gap 



2-105 



= O»o 



B = IL 



V, 



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) 


















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J 
















_ 


jjL 


^ 


K'-o.eo , 

J<''0.70 


-^ 
N, 






















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r 


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60 X 


>v 














is: 


r 


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V k'«o.io 
















/ 






\ 


\ 














/ 








"^ 


\ 







I 2 



8 10 12 14 



16 



18 20 



>/L 



0=15' V; 



B=IL 



2 -i 



V, 



L 



10 



^ 


- 




X 






















^ 
















5 


\ 




















v^ 


k'«0.8 




V 














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.k'"O.S 


















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1 2 



8 



10 12 14 16 18 20 

y^L (Johnson, 1952) 

Figure 2-55. Diffraction for a Breakwater Gap of One Wave 

Length Width ((|) = and 15°) 



2-106 



0=30 



B=IL 




10 









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y 


/ 
















4 




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j<''0,9 


^^^^ 


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s 






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y 














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8 10 12 14 16 18 20 

y/1 



= 45« 



B=IL 



H 



10 



12 





-^ 




V 
















/ 


^ 




) 
















.,^^ 




/ 


/ 


















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ji-^ 












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^t^ 


■ JCT 


s^^ 


1 "r ^-« - K 'O-B 




r 












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^^_ 


/ 


- — 


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V 












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<'»0.2 




y 

































I 2 



8 10 12 14 16 18 20 



y/i 



(Johnson, 1952) 



Figure 2-56. Diffraction for a Breakwater Gap of One 
Wave Length Width {^ = 30 and 45°) 



2-107 




12 4 6 8 10 12 14 16 le 20 



y/L 



07 75" 2 
I 



B=IL ' 

2 



10 



12 



-. 


N, 






^ 


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K 


S 

.1.0 


V 


^ 








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^^ 


^^ 




N^ 


^ 


R<^ 


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\ 




N 












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12 4 6 8 10 12 14 16 18 20 

y/i 



'L (Johnson, 1952) 

Figure 2-57. Diffraction for a Breakwater Gap of One 
Wavelength Width ((j) = 60 and 75°) 



2-108 




( Johnson, 1952) 



Figure 2-58. Diffraction Diagram for a Gap of Two Wave 
Lengths and a 45° Approach Compared with 
That for a Gap Width ^2^ Wave lengths with 
a 90° Approach 



2-109 



diagram shoreward to the breakwater; (b) at this point, constructing a 
diffraction diagram carrying successive crests three or four wavelengths 
shoreward, if possible; and (c) with the wave crest and wave direction 
indicated by the last shoreward wave crest determined from the diffraction 
diagram, constructing a new refraction diagram to the breaker line. The 
work of Mobarek (1962) on the effect of bottom slope on wave diffraction 
indicates that the method presented here is suitable for medium-period 
waves. For long-period waves the effect of shoaling (Section 2.32) should 
be considered. For the condition when the bottom contours are parallel to 
the wave crests, the sloping bottom probably has little effect on diffrac- 
tion. A typical refraction-diffraction diagram and the method for deter- 
mining combined refraction-diffraction coefficients are shown in Figure 
2-59. When a wave crest is not of uniform height, as when a wave is under- 
going refraction, a lateral flow of energy - wave diffraction - will occur 
along the wave crest. Therefore diffraction can occur without the wave 
moving past a structure although the diffraction effects are visually more 
dramatic at the structure. 

2.5 WAVE REFLECTION 

2.51 GENERAL 

Water waves may be either partially or totally reflected from both 
natural and manmade barriers. (See Figure 2-60.) Wave reflection may 
often be as important a consideration as refraction and diffraction in the 
design of coastal structures, particularly for structures associated with 
development of harbors. Reflection of waves implies a reflection of wave 
energy as opposed to energy dissipation. Consequently, multiple reflections 
and absence of sufficient energy dissipation within a harbor complex can 
result in a buildup of energy which appears as wave agitation and surging 
in the harbor. These surface fluctuations may cause excessive motion of 
moored ships and other floating facilities, and result in the development 
of great strains on mooring lines. Therefore seawalls, bulkheads and 
revetments inside of harbors should dissipate rather than reflect incident 
wave energy whenever possible. Natural beaches in a harbor are excellent 
wave energy dissipaters and proposed harbor modifications which would 
decrease beach areas should be carefully evaluated prior to construction. 
Hydraulic model studies are often necessary to evaluate such proposed 
changes. The importance of wave reflection and its effect on harbor 
development are discussed by Bretschneider (1966), Lee (1964), and 
LeMehaute (1965) ; harbor resonance is discussed by Raichlen (1965) . 

A measure of how much a barrier reflects waves is given by the ratio 
of the reflected wave height H^, to the incident wave height H^ which 
is termed the reflection coefficient xJ hence x = H^j/H^. The magnitude 
of X varies from 1.0 for total reflection to for no reflection; how- 
ever, a small value of x does not necessarily imply that wave energy is 
dissipated by a structure since energy may be transmitted through such 
structures as permeable, rubble-mound breakwaters. A transmission co- 
efficient may be defined as the ratio of transmitted wave height H^, to 
incident wave height H^. In general, both the reflection coefficient 



2-110 




Breakwater 



X\\\\\\\\\\V^^^\\\\S\^V\^^^ ^^^^ 



b| Lines of Equal Diffraction Coefficient (k') 



Wave Crests 




Over-oil retroction- diffraction coefficient is given 

by (Kr) (k') A/b^TTj 
Where Kp^Refroction coefficient to breakwater. 
K' ^Diffraction coefficient ot point on 
diffracted wave crest from which 
orthogonolis drawn. 
b^Orthogonal spocing at diffracted wove 

crest 
b =Orthogonol spacing nearer shore. 



Figure 2-59. Single Breakwater - Refraction - Diffraction Combined 



2-III 




December 1952 
Figure 2-60. Wave Reflection at Hamlin Beach, New York 



2-JI2 



and the transmission coefficient will depend upon the geometry and compo- 
sition of a structure and the incident wave characteristics such as wave 
steepness and relative depth d/L, at the structure site. 

2.52 REFLECTION FROM IMPERMEABLE, VERTICAL WALLS (LINEAR THEORY) 

Impermeable vertical walls will reflect almost all incident wave 
energy unless they are fronted by rubble toe protection or are extremely 
rough. The reflection coefficient x is therefore equal to approximately 
1.0 and the height of a reflected wave will be equal to the height of the 
incident wave. Although some experiments with smooth, vertical, impermeable 
walls appear to show a significant decrease of x with increasing wave 
steepness, Domzig (1955), Coda and Abe (1968) have shown that this paradox 
probably results from the experimental technique, based on linear wave 
theory, used to determine x- The use of a higher order theory to describe 
the water motion in front of the wall gives a reflection coefficient of 
1.0 and satisfies the conservation of energy principle. 

Wave motion in front of a perfectly reflecting vertical wall subjected 
to monochromatic waves moving in a direction perpendicular to the barrier 
can be determined by superposing two waves with identical wave niombers, 
periods and amplitudes but traveling in opposite directions. The water 
surface of the incident wave is given to a first order (linear) approxi- 
mation by Equation 2-10, 




(2-10) 



and the reflected wave by. 



H. 



T? = cos 

' 2 



+ 

L T 



Consequently, the water surface is given by the sum of n^ and n^ °^- 
since H^ = H^, 



H, 



T? = 7?,- + T?^ 



/27rx 2jrt\ /27rx 27rt^ 

cos — + cos + 

\L T/ I L T , 



which reduces to 



27rx 2-nt 

■n = H. cos cos -— - 

' L T 



(2-79) 



Equation 2-79 represents the water surface for a standing wave or olapotis 
which is periodic in time and in x having a maximum height of 2H^ when 
both cos(2ttx/L) and cos(2TTt/T) equal 1. The water surface profile as a 
function of 2ttx/L for several values of 27rt/T are shown in Figure 2-61. 
There are some points (nodes) on the profile where the water surface 



2-113 




2-114 



remains at the SWL for all values of t and other points (antinodes) where 
the water particle excursion at the surface, is 2 H^ or twice the incident 
wave height. The equations describing the water particle motion show that 
the velocity is always horizontal under the nodes and always vertical under 
the antinodes. At intermediate points, the water particles move along 
diagonal lines as shown in Figure 2-61. Since water motion at the anti- 
nodes is purely vertical, the presence of a vertical wall at any antinode 
will not change the flow pattern described since there is no flow across 
the vertical barrier and equivalently, there is no flow across a vertical 
line passing through an antinode. (For the linear theory discussion here, 
the water contained between any two antinodes will remain between those 
two antinodes.) Consequently, the flow described here is valid for a 
barrier at 2ttx/L = (x = 0) since there is an antinode at that location. 

2.53 REFLECTIONS IN AN ENCLOSED BASIN 

Some insight can be obtained about the phenomenon of the resonant 
behavior of harbors and other enclosed bodies of water by examining the 
standing wave system previously described. The possible resonant oscillat- 
tions between two vertical walls can be described by locating the two 
barriers so that they are both at antinodes; for example, barriers at 
X = and tt or x = and 2tt, etc. represent possible modes of oscillation. 
If the barriers are taken at x = and x = it, there is one-half of a wave 
in the basin or, if Iq is the basin length, Iq - L/2. Since the wave- 
length is given by Equation 2-4 

gT^ , /27rd\ 
L = ^- tanh — , 
2jr \ L / 

the period of this fundamental mode of oscillation is, 

(2-80) 



T = 



47^4 



gtanh (jrd/^) 



The next possible resonant mode occurs when there is one complete wave in 
the basin (barriers at x = and x = 2ir) and the next mode when there are 
3/2 waves in the basin (barriers at x = and x = 37t/2, etc. In general, 

Jlg = jL/2, where j = 1, 2, In reality, the length of a natural or 

manmade basin 2.^, is fixed and the wavelength of the resonant wave con- 
tained in the basin will be the variable; hence, 

in 

L = — j = 1,2,----, (2-81) 

may be thought of as defining the wavelengths capable of causing resonance 
in a basin of length l^. The general form of Equation 2-80 is found by 



2-115 



substituting Equation 2-81 into the expression for the wavelength; there- 
fore, 

V2 



^J = 



47.^3 



jg tanh (7rjd/4) 



, j = 1,2,---- (2-82) 



For an enclosed harbor, of approximately rectangular planform with length, 
%, waves entering through a breakwater gap having a predominant period 
close to one of those given by Equation 2-82 for small values of j, may 
cause significant agitation unless some effective energy dissipation 
mechanism is present. The addition of energy to the basin at the resonant 
(or excitation) frequency (f^- = 1/Ty) is said to excite the basin. 

Equation 2-82 was developed by assxaming the end boundaries to be 
vertical; however, it is still approximately valid so long as the end 
boundaries remain highly reflective to wave motion. Sloping boundaries, 
such as beaches, while usually effective energy dissipaters, may be signifi- 
cantly reflective if the incident waves are extremely long. The effect of 
sloping boundaries and their reflectivity to waves of differing character- 
istics is given in Section 2.54, Wave Reflection from Beaches. 

Long-period resonant oscillations in large lakes and other large 
enclosed bodies of water are termed seiches. The periods of seiches may 
range from a few minutes up to several hours, depending upon the geometry 
of the particular basin. In general, these basins are shallow with respect 
to their length; hence, tanh (irjd/JJ.g) in Equation 2-82 becomes approximately 
equal to 7rjd/Jlg and 

24 1 

T. = -r — r-w j = 1,2, (small values) . (2-83) 

] j (gd)'^ 

Equation 2-83 is termed Merian's equation. In natural basins, complex 
geometry and variable depth will make the direct application of Equation 
2-83 difficult; however, it may serve as a useful first approximation for 
enclosed basins. For basins open at one end, different modes of oscilla- 
tion exist since resonance will occur when a node is at the open end of 
the basin and the fundamental oscillation occurs when there is one quarter 
of a wave in the basin; hence, ilg' = L/4 for the fundamental mode and 
T = 4£g'//gT; In general ^g' = (2- - l)L/4, and 

44' 1 
'^i = O^ (^^ j = 1, 2, • • • • (small values) . (2-84) 

Note that higher modes occur when there are 3, 5, ...., 2^ - 1, etc., 
quarters of a wave within the basin. 



2-116 



************** EXAMPLE PROBLEM 



************** 



GIVEN: Lake Erie has a mean depth of d = 61 feet and its length % is 
220" miles or 116,160 feet. 

FIND : The fundamental period of oscillation T-, if j = 1. 
SOLUT ION : From Equation 2-83 for an enclosed basin, 

4 1 



211, 
T 



1 



T. = 



2(116,160) 



» 1 [32.2(61)]^^ 



Tj = 52,420 sec. or 14.56 hrs. 

Considering the variability of the actual lake cross-section, this 
result is surprisingly close to the actual observed period of 14.38 
hours (Platzman and Rao, 1963). Such close agreement may not always 
result. 

************************************* 
Note: Additional discussion of seiching is presented in Section 3.84. 



2.54 WAVE REFLECTION FROM BEACHES 

The amount of wave energy reflected from a beach depends upon the 
roughness, permeability and slope of the beach in addition to the steep- 
ness and angle of approach of incident waves. Miche (1951) assumed that 
the reflection coefficient for a beach x> could be described as the 
product of two factors by the expression, 

X = X,X2 , (2-85) 

where x-, depends on the roughness and permeability of the beach and is 
independent of the slope, while Xz depends on the beach slope and the 
wave steepness. 

Based on measurements made by Schoemaker and Thijsse (1949), Miche 
found that x, *» 0-8 for smooth impervious beaches. A value of x * 0.3 



2-117 



to 0.6 has been recommended for rough slopes and step-faced structures 
The second factor 



X2 is given by 



01 o 



01 o 




(2-86a) 



'\L 




(2-86b) 



where (n^/Lo^rnax 



'o I \ o /max 

is constant for a particular beach slope and is given by 



H, 




(2-87) 



in which 3 is the angle the beach makes with the horizontal (tan B = 
beach slope) , and Hq/^q is the incident wave steepness in deep water. 
^^o/^o^max '-^^ ^® considered a cut-off steepness; waves steeper than 
(H^/L^^max will be only partially reflected while waves with a steepness 
less than (Mo/^o^max "iH ^e almost totally reflected. Equation 2-87 is 
given in graphical form in Figure 2-62, and Equation 2-86 is shown in 
Figure 2-63. These equations and figures are valid for impervious beaches 
with waves approaching at a right angle to the beach. 



************** 



EXAMPLE PROBLEM 



*************** 



GIVEN : A wave with a period of T = 10 second, and a deepwater height of 
Hq = 5 feet impinges on an impermeable revetment with a slope of tan 3 
= 0.20. 



FIND : 

(a) Determine the reflection coefficient x. 

(h) What will be the steepest incident wave almost totally reflected 
from the given revetment? 

SOLUTION. Calculate: 



tanj3 = 


0.20; = 5 

tan|3 


Lo = 


gT^ 32.2(100) 

27r In 


Ho 


= 0.00975 = 



512 feet. 



512 



« 0.01 



2-118 




cotangent ^ - 



(ofttr Micht, 1951) 



tangent )S 
Figure 2-62. (Ho/Lo)max Versus Beoch Slope 



X,- 







max 0.5 - 



2 3 4 56789 10 

/.«»/,n«onf fl - I (ofUr Micht, 1951) 

cotangent p - -tt 

tongent p 

Figure 2-63. Xg Versus Beach Slope for Various Values of Hq/Lq 



2-119 




Entering Figure 2-63 with cot B = 5, and using the curve for Ho/ho - 
0.01, a value o£ " X2 ~ 0-41 is found. Assuming that since the beach 

is impermeable, Xi = 0.8 and 

X = X, X2 = 0.8(0.41) = 0.33 . 

The steepest incident wave which will be nearly perfectly reflected from 
the given revetment is, from Figure 2-62, 



0.005 . 



It is interesting to note the effectiveness of flat beaches in dissipat- 
ing wave energy by considering the above wave on a beach having a slope 
of 0.02 (1:50). From Equation 2-87 (noting that 3 « sin B « tan B = 0.02), 

I— = 0.000014. 
\ ol max 
Hence 

X = X, Xj = 0.8(0.0014) = 0.0011, 

or the height of the reflected wave is about 0.1 percent of the incident 
wave height . 

As indicated by the dependence of reflection coefficient on incident 
wave steepness, a beach will selectively dissipate wave energy, dissipat- 
ing the energy of relatively short steep waves and reflecting the energy 
of the longer, flatter waves. 

************************************* 

2.6 BREAKING WAVES 

2.61 DEEP WATER 

The maximum height of a wave travelling in deep water is limited by 
a maximum wave steepness for which the wave form can remain stable. Waves 
reaching the limiting steepness will begin to break and in so doing, will 
dissipate a part of their energy. Based on theoretical considerations, 
Michell (1893) found the limiting steepness to be given by, 

Ho 1 

— = 0.142 =* - . (2-88) 

which occurs when the crest angle as shown in Figure 2-64 is 120 . This 
limiting steepness occurs when the water particle velocity at the wave 
crest just equals the wave celerity; a further increase in steepness would 



2-120 



result in particle velocities at the wave crest greater than the wave 
celerity and, consequently, instability. 




Limiting steepness -p- = 0.142 

Figure 2-64. Wave of Limiting Steepness in Deep Water 

2.62 ■ SHOALING WATER 

When a wave moves into shoaling water, the limiting steepness which 
it can attain decreases, being a function of both the relative depth d/L, 
and the beach slope m, perpendicular to the direction of wave advance. 
A wave of given deepwater characteristics will move toward a shore until 
the water becomes shallow enough to initiate breaking, this depth is 
usually denoted as d]-, and termed the breaking depth. Munk (1949) derived 
several relationships from a modified solitary wave theory relating the 
breaker height H^, the breaking depth d^,, the unrefracted deepwater 
wave height H^, and the deepwater wavelength L^. His expressions are 
given by 






1 



3.3(HyL^)H 



(2-89) 



and 



H. 



= 1.28, 



(2-90) 



The ratio H^/H^ is frequently termed the breaker height index. Subse- 
quent observations and investigations by Iversen (1952, 1953), Galvin 
(1969), and Coda (1970) among others, have established that H2,/H^ and 
dy/U-jj depend on beach slope and on incident wave steepness. Figure 2-65 
shows Coda's empirically derived relationships between H^j/H^ and H^/L^ 
for several beach slopes. Curves shown on the figure are fitted to widely 
scattered data; however they illustrate a dependence of H^j/H^^ on the 
beach slope. Empirical relationships between d^/H^, and H^^/gT^ for 
various beach slopes are presented in Figure 2-66. It is recommended 
that Figures 2-65 and 2-66 be used, rather than Equations 2-89 and 2-90, 
for making estimates of the depth at breaking or the maximum breaker 



2-121 




2-122 



o 

(VJ 













^Sl \. ' ' ^L^L 


■ ' n 






Al_v ^v ^v I^v 










1 








j 






— N \ ' ^ '^ 


i i 




u> 


"v X \ 1 ^ ' 






o 




X i 




^ L \fc^^ ^V 








^ xl I Pv '"' 








— ■ ml\. 1 X 










\ \ 6 






\K N ' \ 


\ \ 






— yV \ : \ 


\ \ 






t\ \ > 


. \ \ 


CO 




V^ \ ; 


\ \\: -- 


(A 




— ~ ' Sk 1 ^ ' 


\l \\ 


Q) 




\\1 1 


1 N ^ \ lo 


c 

Q. 




■" wk Y 


\ >J\ 




«— V * 


I ^ \ ^ - o 




\\ n 


l' S N s • 




ir> 


■" \* j 


N \ \\ 




1 \ V 




(O 


o 


\ 


\' L 1 L I ! 1 




I ' \} 


\ 'L 1 ^1 






X t \ 


\ \ i \l i ^ i ' 






" _j_ _ : " \ 


\ \ ' \ 1 




- 








V li 1 ' ^. * 








a Y Y - 1 - 1 . I 


Q> 




— . 










V '\ N V V o 


U) 






) ^ \i \ ' ^ 






- 


\ * \ V I V 


U> 






1 I \ ' ' 


3 




~ 


ft \ ^ *L I L 


O) 




_ 


ft ' ! > \ \ 


h- 


* 






a> 


o 





W \ 1 " 1 \ \ ^ 


> 


_ 


«\ V' \ k - r^ 










o> 




- 


-jp A^ -s - * - + - ° 


c 














n\ ' • . L " * 


je 




- 


\\y \ \ ^ . ' 


o 






W \ ^ \ 


a> 
















m 






1 \ \ \ \ L * \ ' " <■! 






IJJC X I \ iiV ^ ' ^. ■" 7 


^— 




■ ■ ' "■ 


l\ \ \ \ P X •■„ 


o 












1 


nil \ \ \'\\** 









111 \ I \\(0 


Q. 


CJ) 




111 y ^ \ \\ 




- 


111 '\ \ \ \\ "O 










o 






111 i 1 \ \' ' '\ \l ri 1 ' (S 






— 


ll ' 1 1 V ill 111 ^ 1 J-i 


in 






li i \ \ \'i'\]-"' '' ^O 








II k 1 V \ \ \ *" -^ " 


<u 




_ 


• iV \\ \\\ o 


c 






111 \ \ \ \\'- 












_ - - _ 


\ \ \ \ f— \ \ "^ 








■^ " :t dC \ \a\\0 


c 
a> 






J- : : L . ^ 1 A ^\ ^\ \ c^f 




_ 




CVJ 




\r ir \ O \ '-A \ OC . S 


F 


o 




i- -3-- J\ <^\ -^X-^WzLL _. S 




_ ■ 


_ 1 . ._^\ \ rf\ -X.oVx 4- - • 


n 












_ - 


1 I ! \ i '' \ "^ \ "^X ^ .\ V l_ 


• 






1 "t^\ ' 0\ J^X r^XoWt 


(O 






^\-^\^\ Wt^t 


u> 




— 









\ ' \ -^\ r~ \ /^\ \ \' 








" _. ^'*^\ -^\ o\ \ \ V "T 


OJ 




- 








_ 1 


"*" c;:«^ o\T^\ \n\ o\ \ \\ - o 


a> 








i- 




— 


1 '^"•i'' \ . \ — ^\ '\ I \ x \ o 


3 






' ^\ "1 ''^l ^\ \ \ \\ 












b 


^— < 


1 ''1 /N.l\ ^^ \ \ \ \ I \ 


ll 


L 


1 -^ "~~ 1 t^ I ^^\ \ \ I \ \ 








«, *~^l ^\ ^\ \ \ \ \ \ 






~ 1 ■ 


~J -—1 \ \ \ \ \ \ \ OJ 








-; o\ ^\ \ \ \ \ W o 






— 










~~i^ C>\ \ \ \ \ \ 1 \i\- -f d 






1 








■^ 


rr \ \ \ \ \ \V\' 








T \ \\ \\\\\ 








1 1 \\ \\\\i 






— 


t 1 \\ \ \\\\ 








1 I'V'll 'll \ \\ 








1 ill ^ ' 1 ' 1 1 i \ \ \ 






-f- 
















o 00 to 'J; 
csi — — — 


CM O 00 (D 

^\^ - - ° ° 





2-123 



height in a given depth since the figures take into consideration the 
observed dependence of d^^/H^, and H^j/H^ on beach slope. The curves 
in Figure 2-66 are given by 

_± ^ ^ (2-91) 

Hfc b-(aHj,/gT^) ' 

where a and b are functions of the beach slope m, and may be approxi- 
mated by 

a = 1.36g(l-e-i5'") C2-92) 



b = 



1-56 (2-93) 

(l + e-19.5m) 



Breaking waves have been classified as spilling, plunging or surging 
depending on the way in which they break (Patrick and Wiegel, 1955), and 
(Wiegel, 1964). Spilling breakers break gradually and are characterized 
by white water at the crest. (See Figure '2-67.) Plunging breakers curl 
over at the crest with a plunging forward of the mass of water at the 
crest. (See Figure 2-68.) Surging breakers build up as if to form a 
plunging breaker but the base of the wave surges up the beach before the 
crest can plunge forward. (See Figure 2-69.) Further subdivision of 
breaker types has also been proposed. The term collapsing breaker is 
sometimes used (Galvin, 1968) to describe breakers in the transition from 
plunging to surging. (See Figure 2-70.) In actuality, the transition 
from one breaker type to another is gradual without distinct dividing 
lines; however, Patrick and Wiegel (1955) presented ranges of H^/L^ for 
several beach slopes for which each type of breaker can be expected to 
occur. This information is also presented in Figure 2-65 in the form of 
three regions on the Uj^/^o vs. H^/L^ plane. An example illustrating the 
estimation of breaker parameters follows. 



************** EXAMPLE PROBLEM 



***** 



GIVEN : A beach having a 1:20 slope; a wave with deepwater height of 
H^ = 5 feet and a period of T = 10 seconds. Assume that a refraction 
analysis gives a refraction coefficient, Kff = (bo/b)^/^ = 1.05 at the 
point where breaking is expected to occur. 



FIND : The breaker height Hh and the depth db at which breaking occurs, 
SOLUTION : The unrefracted deepwater height H^ can be found from 

h' / b \'''' 

— = Kp = T- (See Section 2.32), 



2-124 




Figure l-dl . Spilling Breaking Wave 




Figure 2-68. Plunging Breaking Wave 



2-125 




Figure 2-69. Surging Breaking Wave 




Figure 2-70. Collapsing Breaking Wave 



2-126 



hence, 

H^ = 1.05(5) = 5.25 feet, 
and since, L^ = 5,12T^ (linear wave theory). 



% 5.25 



L 5.12(10)^ 



= 0.010 



o 



From Figure 2-65, entering with H^/L^ = 0.010 and intersecting the curve 
for a slope of 1:20 (m = 0.05) results in H^/H^ = 1.65. Therefore 




H^ = 1.65(5.25) = 8.66 feet . 
To determine the depth at breaking calculate: 



^b 8.66 



gT^ 32.2(10)2 
and enter Figure 2-66 for m = 0.050. 



0.00269 , 



^b 



= 0.90 



Thus djy = 0.90 (8.66) =7.80 feet, and therefore the wave will break 
when it is approximately 7.80/(0.05) = 156 feet from the shoreline, 
assuming a constant nearshore slope. The initial value selected for 
the refraction coefficient should now be checked to determine if it is 
correct for the actual breaker location as found in the. solution. If 
necessary, a corrected value for the refraction coefficient should be 
used and the breaker height recomputed. The example wave will result 
in a plunging breaker. (See Figure 2-65.) 



2-127 



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2-129 



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2-130 



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2-131 



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2-133 



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2-134 



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2-135 



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2-136 



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2-137 



CHAPTER 3 



WAVE AND 



WATER LEVEL 



PREDICTIONS 




OCEAN CITY, NEW JERSEY - 9 March 1962 



CHAPTER 3 
WAVE AND WATER-LEVEL PREDICTIONS 

3.1 INTRODUCTION 

Chapter 2, treated phenomena associated with surface waves as though 
each phenomenon could be considered separately without regard to other 
phenomena. Surface waves were discussed from the standpoint of motions 
and transformations without regard to wave generation. Furthermore, the 
water level, stiltwater level (SWL) , on which the waves propagated was 
assumed known. 

In this chapter, wave observations are presented to show characteris- 
tics of surface waves in nature. The characteristics of real waves are 
much less regular than those implied by theory. Also presented are pro- 
cedures for representing the complexity of the real sea by a small number 
of parameters. Deviations between the actual waves and the parameter 
values are discussed. 

Theory for wave generation is reviewed to show progress in explaining 
and predicting the actual complexity of the sea. Wave prediction is 
called hindaasting when based on past meteorological conditions, and fore- 
casting when based on predicted conditions. The same procedures are used 
for hindcasting and forecasting; the only difference is the source of 
meteorological data. The most advanced prediction techniques currently 
available can be used only in a few laboratories, because of the need for 
electronic computers, the sophistication of the models, and the need for 
correct weather data. However, simplified wave prediction techniques, 
suitable for use by field offices or a design group are presented. 

While simplified prediction systems will not solve all problems, they 
can be used to indicate probable wave conditions for most design studies. 
Simplified wave prediction can also be used to obtain statistical wave 
data over several years. 

Review of prediction theories is presented to give the reader more 
perspective for the simplified prediction methods that are presently 
available. This will justify confidence in some applications of the 
simplified procedures, will aid in recognizing unexpected difficulties 
when they occur, and will indicate some conditions in which they are not 
adequate . 

The graphs in Sections 3.5, Simplified Wave Prediction Models, and 
3.6, Wave Forecasting for Shallow Water Areas, may be read with the pre- 
cision justified by the underlying theory. The equations were derived 
originally from graphs, and do not provide any physical understanding. 
Calculations with the graphs should be carried out to tenths or hundredths 
where this is feasible, and then rounded off in the final result. 

Predictions are compared with available observations wherever possible 
to indicate their accuracy. Calibration of techniques applied to a spe- 
cific geographic area by comparison with available observations is always 
desirable. 



3- 



The problem of obtaining wind information for wave hindcasting is 
discussed, and specific instructions for estimating wind parameters are 
given. 

Water levels continuously change. Changes due to astronomical tides 
are predictable, and are well documented for many areas. Fluctuations due 
to meteorological conditions are not as predictable, and are less well 
documented. 

Many factors govern water levels at a shore during a storm. The 
principle factor is the effect of wind blowing over water. Some of the 
largest increases in water level are due to severe storms, such as hurri- 
canes, which can cause storm surges higher than 25 feet at some locations 
on the open coast and even higher water levels in bays and estuaries. 
Estimating water levels caused by meteorological conditions is complex, 
even for the simplest cases; and unfortunately, the best approaches avail- 
able for predicting these water levels are elaborate computational tech- 
niques which require the use of large digital computers. 

3.2 CHARACTERISTICS OF OCEAN WAVES 

The earlier discussion of waves was concerned with idealized, mono- 
chromatic waves. The pattern of waves on any body of water exposed to 
atmospheric winds generally contains waves of many periods. Typical 
records from a recording gage during periods of steep waves (Fig. 3-1) 
indicate that heights and periods of real waves are not as constant as is 
assumed in theory. Wavelengths and directions of propagation are also 
variable. (See Figure 3-2.) The prototype is so complex that some 
idealization is required. 

3.21 SIGNIFICANT WAVE HEIGHT AND PERIOD 

An early idealized description of ocean waves postulated a significant 
height and significant period, that would represent the characteristics of 
the real sea in the form of monochromatic waves. 

The representation of a wave field by significant height and period 
has the advantage of retaining much of the insight gained from the theo- 
retical studies. Its value has been demonstrated in the solution of 
engineering problems. For some problems this representation appears 
adequate; for others it is useful, but not entirely satisfactory. 

To apply the significant wave concept it is necessary to define the 
height and period parameters from wave observations. Munk (1944) defined 
significant wave height, as the average height of the one-third highest 
waves, and stated that it was about equal to the average height of the 
waves as estimated by an experienced observer. This definition, while 
useful, has some drawbacks in wave-record analysis. It is not always 
clear which irregularities in the wave record should be counted to deter- 
mine the total number of waves on which to compute the average height of 
the one-third highest. The significant wave height is written as H1/3 or 
simply Hg. 



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3-4 



The significant wave period obtained by visual observations of waves 
is likely to be the average period of 10 to 15 successive prominent waves. 
When determined from gage records, the significant period is apt to be the 
average period of the subjectively estimated most prominent waves, or the 
average period of all waves whose troughs are below and whose crests are 
above the mean water level, (zero up crossing method). 

3.22 WAVE HEIGHT VARIABILITY 

When the heights of individual waves on a wave record are ranked from 
the highest to lowest, the frequency of occurrence of waves above any given 
value is given to a close approximation by the cumulative form of the 
Rayleigh distribution. This fact can be used to estimate the average 
height of the one-third highest waves from measurements of a few of the 
highest waves, or to estimate the height of a wave of any arbitrary 
frequency from a knowledge of the significant wave height. According to 
the Rayleigh distribution function, the probability that the wave height 
H is more than some arbitrary value of H referred to as H is given 
by 

P(H > H) = e ^ '""' (3-1) 

where Hmis is a parameter^of the distribution, and P(H > H) is the number 
n of waves larger than H divided by the total number N of waves in 
the record. Thus P has the form n/N. The value Hrms is called the 
root-mean- square height and is defined by 




^rm. = V. .2. H • (3-2) 



It was shown in Section 2.238, Wave Energy and Power, that the total energy 
per unit surface area is given by 



E = '-fil 



The average energy per unit surface area for a number of waves is given by 



Pg 1 



N 



where Hj is the height of successive individual waves, and (¥)^ is the 
average energy per unit surface area of all waves considered. Thus H^ms 
is a measure of average wave energy. Calculation of Hj^g by Equation 3-2 
is somewhat less subjective than direct evaluation of the Hg because more 
emphasis is placed on the larger, better defined waves. The calculation 
can be made more objective by substitution of n/N for P(H > iT) in 
Equation 3-1 and taking natural logarithms of both sides to obtain 

Ln(n) = Ln(N) - (H;^pH^ . (3_4) 

3-5 



By making the substitutions 

y(n) = Ln(n), a = Ln(N), b = - H;^^, x(n) = H'(n) . 

Equation 3-4 may be written as 

y(n) = a + bx(n) . (3-5) 

The constants a and b can be found graphically or by fitting a least- 
square regression line to the observations. The parameters N and Harris 
may be computed from a and b. The value of N found in this way, is 
the value that provides the best fit between the observed distribution of 
identified waves and the Rayleigh distribution function. It is generally 
a little larger than the number of waves actually identified in the record. 
This seems reasonable because some very small waves are generally neglected 
in interpreting the record. When the observed wave heights are scaled by 
^rms) that is, made dimensionless by dividing each observed height by 
^rms > then data from all observations may be combined into a single plot. 
Points from scaled 15-minute samples are superimposed on Figure 3-3 to 
show the scatter to be expected from analyzing individual observations in 
this manner. 

Data from 72 scaled 15-minute samples representing 11,678 observed 
waves have been combined in this manner to produce Figure 3-4. The theo- 
retical height appears to be about 5 percent greater than the observed 
height for a probability of 0.01 and 15 percent at a probability of 0.0001. 
It is possible that the difference between the actual and theoretical 
heights of highest waves is due to breaking of the very highest waves 
before they reach the coastal wave gages. 

Equation 3-1 can be established rigorously for restrictive conditions, 
and empirically for a much wider range of conditions. If Equation 3-1 is 
accepted as an exact law, the probability density function can be obtained 
in the form 

f[(H-AH) < H < (fi+ AH)] = l^—\ He ^^'""' . (3-6) 




The height of the wave with any given probability n/N of being exceeded 
may be determined approximately from curve a in Figure 3-5 or from the 
equation. 







(3-7) 



3-6 



0.000 



0.0005 



0.001 



0.005 



~ 0.0 



O 

o 



a> 

> 



3 

E 

S 




0.05 



From fifteen 15-minute records 
containing a total of 2,342 waves 
(3,007 waves calculated) 



(See discussion below Equation 3-8) 
_l I I I I 



1.0 1.2 



1.4 1.6 
1.42 



1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 

Scaled Heigtit (H/Hrms) 

Figure 3-3. Theoretical and Observed Wave-Height Distributions. 
Observed distributions for 15 individual IS-minute 
observations from several Atlantic coast wave gages 
are superimposed on the Rayleigh distribution curve 



3-7 



0.000 



0.0005 



0.001 



Q. 0.005 



- 0.0 



o 
.a 
o 



a> 

> 



3 

E 




0.05 



0.5 



From seventy-two 15-minute samples 
containing a total of 11,678 waves 
(15,364 calculated waves) 



Hg Theory 



J__L 



1.0 1.2 1.4 I 
1.42 




2.0 2.2 



2.4 

A 



2.6 



2.8 



3.0 



3.2 



Scaled Heigtit (H/Hrms) 



Figure 3-4. Theoretical and Observed Wave-Height Distributions. 
Observed waves from 72 individual 15-minute 
observations from several Atlantic coast wave gages 
are superimposed on the Rayleigh distribution curve 



3-8 





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{N/u = d) '^ilMqoqoJd aAijDinujno 



3-9 



The average height of all waves with heights greater than H (H) can be 
obtained from the equation. 



/ h' e ' "^'"^ ' d H 



H(H) = 



H 



/ H eVHrmJ dH 



(3-8) 



H 



or from curve b in Figure 3-5. By setting H = 0, all waves are con- 
sidered, and it is found that the average wave height is given by 



H = 0.886 H 



rms ' 



and the significant wave height is given by 

H = 1.416H „, « vTh 



(3-9) 



(3-10) 



In the analysis system used by CERC from 1960 to 1970/ and whenever 
digital recordings cannot be used, the average period of a few of the best 
formed waves is selected as the significant wave period. An estimated 
number of equivalent waves in the record is obtained by dividing the 
duration of the record by this significant period. The highest waves 
are then ranked in order with the highest wave ranked 1. The height of 
the wave ranked nearest 0.135 times the total number of waves is taken as 
the significant wave height. The derivation of this technique is based on 
the assumption that the Rayleigh distribution law is exact. Harris (1970) 
showed that this procedure agrees closely with values obtained by more 
rigorous procedures which require the use of a computer. These procedures 
are described in Section 3.23, Energy Spectra of Waves. 

The following problem illustrates the use of the theoretical wave 
height distribution curves given in Figure 3-5. 

************** EXAMPLE PROBLEM ************** 

GIVEN : Based on an analysis of wave records at a coastal location, the 
significant wave height Hg was estimated to be 10 feet. 

FIND : 

(a) Hjo (Average of the highest 10 percent of all waves) 

(b) H-^ (Average of the highest 1 percent of all waves) 



3-10 



SOLUTION : 1? = Hg = 10 feet 



Using Equation 3-10 



or 



H = 1.416 H^, 
s rms 



H .-1!^ = -15-. 7.06 ft 

'■'"^ 1.416 1.416 



(a) From Figure 3-5, curve b, it is seen that for P = 0.1 (10 percent) 
^^ « 1.80; Hjo = 1.80 H = 1.80(7.06) = 12.7 feet, say 13 feet. 



rms 



(b) Similarly, for P = 0.01 (1 percent) 

2.36; H^ = 2.36 H^^^ = 2.36 (7.06) = 16.7 ft., say 17 ft. 



"/ 



rtns 



Note that; 



^10 12.7 



H 10 ^<^ 



or H,„ = 1.27 H. 



and 



$ 



^ _ 1^ 
H, ~ 10 



or H^ = 1.67 H^ 



************************************* 

Goodknight and Russell (1963) analyzed wave-gage observations taken 
on an oil platform in the Gulf of Mexico during four hurricanes. They 
found agreement adequate for engineering_application between such important 
parameters as Hg, Hio, ihiax, ^rms^ and H, although they did not find 
consistently good agreement between measured wave-height distributions 
and the entire Rayleigh distribution. Borgman (1972) substantiates this 
conclusion from wave observations from other hurricanes. These findings 
are consistent with Figures 3-3, and 3-4, based on wave records recently 
obtained by CERC from shore-based gages. The CERC data include waves from 
both extratropical storms and hurricanes. 

3.23 ENERGY SPECTRA OF WAVES 

The significant wave analysis, although simple in concept, is diffi- 
cult to do objectively, and does not provide all of the information needed 
in engineering design. 



It appears from Figure 3-1 that the wave field might be better de- 
scribed by a sum of sinusoidal terms. That is, the curves in Figure 3-1 
might be better represented by expressions of the type 

N 
77(t)= S a. cos(w.-0.) , (3-11) 

j= 1 J J ' 

where n(t) is the departure of the water surface from its average posi- 
tion as a function of time, aj is the amplitude, toj is the frequency, 
and (jjj is the phase of the j*^ wave at the time t = 0. The values 
of oj are arbitrary, and cj may be assigned any value within suitable 
limits. In analyzing records, however, it is convenient to set 
(iijj - ZiTJ/D, where j is an integer and D is the duration of the obser- 
vation. The dij will be large only for those w^- that are prominent in 
the record. When analyzed in this manner, the significant period may be 
defined as D/j , where j is the value of j corresponding to the 
largest a^-. 

It was shown by Kinsman (1965) , that the average energy of the wave 
train is proportional to the average value of [n(t)]^. This is identical 
to a^ where o is the standard deviation of the wave record. It can 
also be shown that 

1 ^ 
o' = - S a?. (3-12) 

Experimental results and calculations based on the Rayleigh distri- 
bution function show that the significant wave height is approximately 
equal to 4a. Thus, recalling that 



and 



then 



". '^ v/2 H^^^ 



H^ « 4a 



or 



a == 0.25 \/2^H^^^, (3-13) 



^rms * 2 V2' o. (3-14) 

The a? may be regarded as approximations to the energy spectrum 



function E(oj) where 



a? 



E(cj) Aw = -^ . (3-15) 



3-12 



Thus 






E(co)da; . (3-16) 



The spectrum E((jj) permits one to assign specific portions of the 
total wave energy to specific frequency intervals, to recognize that two 
or more periods may be important in describing the wave field, and to give 
an indication of their relative importance. This also permits a first 
approximation to the calculation of velocities and accelerations from a 
record of the wave height in a complex wave field. Several energy spectra, 
computed from coastal zone wave records obtained by CERC, are shown in 
Figure 3-6. 

The international standard unit for frequency measure is the hertz^ 
defined as one cycle per second. The units aycZes pev second and radians 
per second are also widely used. One hertz = 2it radians per second. 

3.24 DIRECTIONAL SPECTRA OF WAVES 

A more complete description of the wave field is required to recognize 
that not all waves are traveling in the same direction. This may be 
written as , 

Tj (x, y, t) = 2 a. cos [co-t - 0- - k (x cos fly + y sin 6-^ , (3- 17) 

where k = Ztt/L, and 9^- is the angle between the x axis and the 
direction of wave propagation, and ^a is the phase of the j^^ wave 
at t = 0. The energy density E(e,u) represents the concentration of 
energy at a particular wave direction 6 , and frequency w , therefore 
the total energy is obtained by integrating E(e,{jo) over all directions 
and frequencies, thus 

27r oo 

E(0,a))dwd9 . (3-18) 



= // 



The concept of directional wave spectra is essential for advanced 
wave-prediction models, but technology has not yet reached the point where 
directional spectra can be routinely recorded or used in engineering design 
studies. Therefore, directional wave spectra are not discussed extensively 
here. 



3-13 



+ 0.4 



iiJ 

e 
> 
o 

o 
o 



+ 0.2 — 



0.0 



OjO 



Period (Seconds) 

2.5 10 




0.2 



0.4 0-0 

Frequency (Hertz) 
Savannah Coast Guard Light Tower 



2.5 




2 



0.4 



0.4 



10 



liJ 



> 
o 



o 
I- 



0.2 



0.0 ^-« 

0.0 



Period (Seconds) 

2.5 10 



T = 10.8 Sec 



H, 



6.9 Ft 




Frequency (Hertz) 



Nags Head, N.C. 
Figure 3-6. 



2.5 




0.2 



4 



Typical Wave Spectra from the Atlantic Coast. 
The ordinate scale is the fraction of total 
wave energy- in each frequency band of 0.011 
Hertz, (one Hertz is one cycle/second). A 
linear frequency scale is shown at the bottom 
of each graph and a nonlinear period scale 
at the top of each graph. 



3-14 



3.3 WA\"E FIELD 

3.31 DBTLOPMEJiT OF k MkVE FIELD 

Various descriptions of the aechanisa of wave g^ieratioo by vind have 
been given, and significant progress in explaining the aechanisa nas re- 
ported by Miles C1957) and Phillips (19S7) . Inteir^r-: : ' ; cnssions of the 
results of aany of the aore pr«»inrait descripticr.s :f i -er.eraticHi by 
wind are given by Kinsaan (1965), Phillips (1966), "i I 1971). 






Laboratory studies, (Hidy and Plate, 1966^ ^-i S-tzci:. i 
1966), carefully designed to Batch the assusipti:-.5 r=.de by Miles and by 
Phillips show reasonably good arrTe-^-" -:'- -'- z - rtretical rrefirtir::- . 
SusBaries of various filed studies .- . -i 1> . ->^7) deacr^trate "ir.a": 
theory provides a reasonable frirt-:r- f:r -.r liI sis of observations. 

The Nliles-Riillips theor>- as t -.ttiti ^- i i:m.-.i- :. t;;-eriBental 
data pemits the foiBulation of a. differential equzticfa governing tie 
growth of wave energ>-. This equat::- e written in a variety of ways. 
(Inoue, 1966, 1967) and Bamett , 1.-: is approadi will cot be disr^s: 
in detail because it requires = itrzi lii^zity co^pater i- i -:rT i-i.rz- 
logical data than is likely to :e f:_-z r ztt": i- = -a;;r f:rT:iit :e--er. 

A brief discussion of the physical concepi; tiloj-ed in tie ci-r-ter 
wave forecast, however, is presented to show tT rtcfings a- i -Trits 
of simpler procedures that can be used in wave irT.asting. 

Growth and dissipation of wave energ)- are very sensitive to wave 
frequency and wave direction relative to the wind direction. Ihus it is 
desirable to consider each narrow band of directi(ms and frequencies 
separately. A change in wave energy depends on the advecticn of energy 
into and out of a region; trans forBatioo of the wind's kinetic energy 
into the energ>- of water waves; dissipation of wave energy into tmbaleiice 
and by fricticm, \-iscosity and breaking; and trans foraatirr. cf vave energy 
at one frequency into wave energ>- at other frequencies. 

Mave energy- is discussed in Section 2.258. Wave Energj- and Power. 
-Althou^ it is known that ener5>' transfers from one band of wave frequen- 
cies to another do take place, this trocess is secondary- to the transfer 
of energ>- froa the atBosx)here to the sea, and is not yet well enon^ onder- 
stood to justify its consideration in a practical wave prediction sdieae. 

Riillips (1957) shewed that the turbulence assrciated with thf fl:- 
of wind near the water v.c>uld create traveling pressure pulses. Thtir 
pulses generate waves traveling at a speed appropriate to the diaensirn; 
of the pressure pulse. Wave grc«t;h ry t-i5 7r::e55 i; - - f- 
the waves are short and ^liien their speed is luenticu. _i-. : . .;: t " 
of the wind velocity in the directirn cf wave travel. The err in; 1 ii:a 
analyzed by Inoue (1966, 1967) indicate- -.'- \' - r effect of : r: lei 
pressure pulses is real, but is only ii:_i ;r. -i eniieth as _^rge ^s t^ie 
original theor>' indicated. 



3-15 



Miles (1957) showed that the waves on the sea surface must be 
matched by waves on the bottom surface of the atmosphere. The speed 
of air and water must be equal at the water surface. Under most meteoro- 
logical conditions, the air speed increases from near to 60 - 90 percent 
of the free air value within 66 feet (20 meters) of the water surface. 
Within a shear zone of this type, energy is extracted from the mean flow 
of the wind and transferred to the waves. The magnitude of this transfer 
at any frequency is proportional to the wave energy already present at 
that frequency. Growth is normally most rapid at high frequencies. The 
energy transfer is also a complex function of the wind profile, the 
turbulence of the air stream, and the vector difference between wind and 
wave velocities. 

The theories of Miles and Phillips predict that waves grow most 
rapidly when the component of the wind speed in the direction of wave 
propagation is equal to the speed of wave propagation. 

The wave generation process discussed by Phillips is very sensitive 
to the structure of the turbulence. This is affected significantly by 
any existing waves, and the temperature gradient in the air near the 
water surface. The turbulence structure in an offshore wind is also 
affected by land surface roughness near the shore. 

The wave generation process discussed by Miles is very sensitive to 
the vertical profile of the wind. This is determined largely by turbulence 
in the wind stream, the temperature profile in the air, and by the rough- 
ness of the sea surface. 

Shorter waves grow most rapidly. Those waves which propagate obliquely 
to the wind are favored, for they are better matched to the component of 
the wind velocity in the direction of wave propagation than those moving 
parallel to the wind. Thus, the first wave pattern to appear for short 
fetches and durations consists of two wave trains forming a rhombic 
pattern with one diagonal along the direction of the mean wind. 

There is a limit to the steepness to which a wave can grow without 
breaking. Shorter waves reach their limiting growth rather quickly; 
longer waves, which grow more slowly but can obtain greater heights, 
then become more prominent. Thus, the apparent direction of propagation 
of the two wave trains tends to coalesce with increasing fetch and duration. 
The length of the region in which a rhombic pattern is apparent may extend 
from a few meters to a few kilometers depending on the width of the basin, 
the wind speed, and previously existing waves. 

Wave growth is significantly affected by any preexisting waves. The 
empirical data analyzed by Inoue (1966, 1967) indicated that the magnitude 
of the effect of seas already present is about eight times the value given 
in the original Miles (1957) theory. Neglecting this effect in early wave 
prediction theories has led to large errors in computing the duration 
required for a fully arisen sea. There are many situations in which the 
largest waves and the waves growing most rapidly are not being propagated 
in the wind direction. 



3-16 



3.32 VERIFICATION OF WAVE HINDCASTING 

Inoue (1967) prepared hindcasts for Weather Station J (located near 
53°N, 18°W) , for the period 15-28 December, 1959, using a differential 
equation embodying the Miles-Phillips theory to predict wave growth. A 
comparison of significant wave heights from shipboard observations and by 
hindcasting at two separate locations near the weather ships is shown in 
Figure 3-7. The location of Ocean Weather Ship J, the mesh points used in 
the numerical calculations, and four other locations discussed below are 
are shown in Figure 3-8. The calculations required meteorological data 
from 519 grid points over the Atlantic Ocean as shown in Figure 3-8. The 
agreement between observed and computed values seems to justify a high 
level of confidence in the basic prediction model. Observed meteorological 
data were interpolated in time and space to provide the required data, 
thus these predictions were hindcasts. 

Bunting and Moskowitz (1970) and Bunting (1970) have compared fore- 
cast wave heights with observations, using the same model with comparable 
results. 

By 1970, it was generally believed that the major remaining difficulty 
in wind wave prediction was the determination of the surface wind field over 
the ocean. (Pore and Richardson, 1967), and (Bunting, 1970). It is partly 
because of the difficulty in obtaining a satisfactory specification of the 
wind field over the sea that simpler wave prediction systems are still 
being used operationally. (Pore and Richardson, 1969), (Shields and Burdwell, 
1970), and (Francis, 1971.) 

3.33 DECAY OF A WAVE FIELD 

Wind-energy can be transferred directly to the waves only when the 
component of the surface wind in the direction of wave travel exceeds the 
speed of wave propagation. Winds may decrease in intensity, pass over 
land, or change in direction to such an extent that wave generation ceases, 
or the waves may propagate out of the generation area. When any of these 
events occurs, the wave field begins to decay. Wave energy travels at a 
speed which increases with the wave period. Thus the energy packet leaving 
the generating area spreads out over a larger area with increasing time. 
The apparent period at the energy front increases and the wave height 
decreases. If the winds subside before the sea is fully arisen, the 
longer waves may begin to decay while the shorter waves are still growing. 
This possibility is recognized in advanced wave prediction techniques. 
The hindcast spectra, computed by the Inoue (1967) model and published by 
Guthrie (1971) show many examples of this for low swell, as do the aerial 
photographs and spectra given by Harris (1971). (See Figures 3-2 and 3-6.) 
This swell is frequently overlooked in visual observations and even in the 
subjective analysis of pen and ink records from coastal wave gages. 

Most coastal areas of the United States are so situated that most of 
the waves reaching them are generated in water so deep that depth has no 
effect on wave generation. In many of these areas, wave characteristics 



3-17 




o 



o 


in 


O in O 


* 


to 


lO CM CNJ 



3-18 



77^ 



. • \ • 


• • • • 


<%^ . A 

> * T..— -—"^ — i" 

» • • • • 1 


• * *\ ' 


• * * * \ 
• • * * * \ 

•••*** A- 

• •••** \ 

• ••••* 1 
• •••** * 


O 


^' <^: 


• • • • 







• ••••• * 
















• • y . 


• • • • 


> • • • • 

• • . .o 

o 


* • • • ,o 


• •••••• 

•••••• , 


o 

^~- — in 


/I 1/1 /' •""^-^.^ ■ »/ 

f/' ' • •» 


' ' • f 

*i'¥ ■ : 


1 '^ 

' , k 

""'"\* * • • • / 

* • • -^"T^s 

• • . / •'"----^ 



C/D 



.E Ph 



A. 



<9. 



3-19 



may be determined by first analyzing meteorological data to find deep- 
water conditions. Then, by analyzing refraction (Section 2.32, General - 
Refraction by Bathymetry.), the changes in wave characteristics as the 
wave moves through shallow water to the shore may be found. In other 
areas, in particular along the North- Atlantic coast, wKere the bathymetry 
is complex, refraction procedure results are frequently difficult to inter- 
pret, and the conversion of deepwater wave data to shallow-water and near- 
shore data becomes laborious and sometimes inaccurate. 

Along the Gulf coast and in many inland lakes, generation of waves by 
wind is appreciably affected by water depth. In addition, the nature and 
extent of transitional and shallow-water regions complicate ordinary re- 
fraction analysis by introducing a bottom-friction factor and associated 
wave energy dissipation. 

3.4 WIND INFORMATION NEEDED FOR WAVE PREDICTION 

Wave prediction from first principles, as described above, requires 
very detailed specification of the wind field near the water surface. This 
is generally developed in two steps: (1) Estimation of the mean fvee air 
wind speed and direction, (This step may be omitted for reservoirs and small 
lakes if surface wind observations are available.), and (2) Estimation of 
the mean surfaae wind speed and direction. 

When the full wave generation process is considered, a large capacity 
computer must be used for the calculations, and fairly complex procedures 
may be used for determining the wind field. Engineers who require wave 
hindcasts for only a few locations, and perhaps for only a few dates must 
employ simpler techniques. A brief discussion of the processes involved 
in determining the surface wind and techniques suitable for use in deter- 
mining the characteristics of the wind field needed for the simplified wave 
prediction model described in Section 3.5, Simplified Wave Prediction Models, 
are given in this section. These procedures will be accurate (within 20 
percent) about two-thirds of the time. The following discussion provides 
guidance for recognizing cases in which the simplified procedures are not 
appropriate. Errors resulting from disregarding the exceptional situations 
tend to be random. Thus climatological summaries, based on hindcast data, 
may be much more accurate than the individual values that go into them. 

Wind reports from ships at sea are generally estimates based on the 
appearance of the waves, the drifting of smoke, or the flapping of flags - 
although some are anemometer measurements. Actually, even if all ships 
were equipped with several aneometers, the wind field over the sea would 
still not be known in sufficient detail or precision to permit full 
exploitation of modem theories for wave generation. 

Fortunately, estimates of the surface wind field that are usefully 
accurate most of the time can be based on the isobaric pattern of synoptic 
weather charts . 



3-20 



Horizontal pressure gradients arise in the atmosphere primarily 
because of density differences, which in turn are generated primarily by 
temperature differences. Wind results from nature's efforts to eliminate 
the pressure gradients, but is modified by many other factors. 

The pressure gradient is nearly always in approximate equilibrium with 
the acceleration produced by the rotation of the earth. The geostrophia 
wind is defined by assuming that exact equilibrium exists, and is given by 

1 dp 
U = -T / . (3-19) 

* pt an 

where U^ is the wind speed, p^ the density of the air, f the coriolis 
parameter, f = 2w sine}), where to = 7.292 x 10"^ radians/second and (f) 
is the latitude, and dp/dn is the horizontal gradient of atmospheric 
pressure. A graphic solution of this equation is given in Figure 3-11, 
Section 3.41, Estimating the Wind Characteristics. The geostrophic wind 
blows parallel to the isobars with low pressure to the left, when looking 
in the direction toward which the wind is blowing, in the Northern 
Hemisphere, and low pressure to the right in the Southern Hemisphere. 
Geostrophic wind is usually the best simple estimate of the true wind in 
the free atmosphere. 

When the trajectories of air particles are curved, equilibriiim wind 
speed is called gradient wind. Gradient wind is stronger than geostrophic 
wind for flow around a high pressure area, and weaker than geostrophic wind 
for flow around low pressure. The magnitude of the difference between 
geostrophic and gradient winds is determined by the curvature of the tra- 
jectories. If the pressure pattern does not change with time and friction 
is neglected, trajectories are parallel with the isobars. The isobar curva- 
ture can be measured from a single weather map, but at least two maps must 
be used to estimate trajectory curvature. There is a tendency by some 
analysts to equate the isobars and trajectories at all times, and to 
compute the gradient wind correction from the isobar curvature. When the 
curvature is small, but the pressure is changing, this tendency may lead 
to incorrect adjustments. Corrections to the geostrophic wind that can- 
not be determined from a single weather map are usually neglected, even 
though they may be more important than the isobaric curvature effect. 

The equilibrium state is further disturbed near the surface of the 
earth by friction. Friction causes the wind to cross the isobars toward 
low pressure at a speed lower than the wind speed in the free air. Over 
water, the average surface wind speed is generally about 60 to 75 percent 
of the free air value, and wind crosses the isobars at an angle of 10 to 
20 degrees. In individual situations, the magnitude of the ratio between 
the surface wind speed and the computed free air speed may vary from 20 to 
more than 100 percent, and the crossing angle may vary from 0° to more than 
90°. The magnitude of these changes is determined by the vertical tempera- 
ture profile and the turbulent viscosity in the atmosphere. 

3-21 



3.41 ESTIMATING THE WIND CHARACTERISTICS 

To predict wave properties from meteorological data by any of the 
simplified techniques, it is necessary to: 

(a) Estimate the mean surface wind speed and direction, as dis- 
cussed in Section 3.4, Wind Information Needed for Wave Prediction; 

(b) delineate a fetch over which the wind is reasonably constant 
in speed and direction, and measure the fetch length, and 

(c) estimate wind duration over the fetch. 

These determinations may be made in many ways depending on the loca- 
tion and the type of meterological data available. For restricted bodies 
of water, such as lakes, the fetch length is often the distance from the 
forecasting point to the opposite shore measured along the wind direction. 
There is no decay distance, and it is often possible to use observational 
data to determine wind speeds and durations. 

When forecasting for oceans or other large bodies of water, the most 
common form of meteorological data used is the synoptic surface weather 
chart. (^Synoptic means that the charts are drawn by analysis of many 
individual items of meteorological data obtained simultaneously over a 
wide area.) These charts depict lines of equal atmospheric pressure, 
called isobars. Wind estimates at sea, based on an analysis of the sea- 
level atmoshperic pressure are generally more reliable than wind observa- 
tions because pressure, unlike wind, can be measured accurately on a 
moving ship. Pressures are recorded in millibars, 1,000 dynes per square 
centimeter. One thousand millibars (a bar) equals 29.53 inches of mercury 
and is 98.7 percent of normal atmoshperic pressure. 

A simplified surface chart for the Pacific Ocean is shown in Figure 
3-9, which is drawn for 27 October 1950 at 0030Z (0030 Greenwich mean 
time). Note the area labelled L in the right center of the chart, and 
the area labelled H in the lower left corner of the chart. These are 
low- and high-pressure areas; the pressures increase moving out from L 
(isobars 972, 975, etc.) and decrease moving out from H (isobars 1026, 
1023, etc.). 

Scattered about the chart are small arrow shafts with a varying num- 
ber of feathers or barbs. The direction of a shaft shows the direction 
of the wind; each one-half feather represents a unit of 5 knots (2.5 
meters/second) in wind speed. Thus, in Figure 3-9 near the point 35°N. 
latitude, 135°W. longitude, there are three such arrows, two with 5h 
feathers which indicate a wind force of 31 to 35 knots (15 to 17.5 meters/ 
second) , and one with 3 feathers indicating a force of 26 to 30 knots 
(13 to 15 meters/second) . 

On an actual chart, much more meteorological data than wind speed and 
direction are shown for each station. This is accomplished by the use of 



3-22 




o 

LO 



U 

(U 
£i 

O 
■t-> 

O 
O 

CNJ 



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tn 
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(U 
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3-23 



coded symbols, letters, and numbers placed at definite points in relation 
to the station dot. A sample model report, showing the amount of informa- 
tion possible to report on a chart, is shown in Figure 3-10. Not all of 
the data shown on this plot are included in each report, and not all of 
the data in the report are plotted on each map. 

Figure 3-11 may be used to facilitate computation of the geostrophic 
wind speed. A measure of the average pressure gradient over the area is 
required. Most synoptic charts are drawn with either a 3- or 4-millibar 
spacing. Sometimes when isobars are crowded, intermediate isobars are 
omitted. Either of these standard spacings is adequate as a measure of 
the geographical distance between isobars. Using Figure 3-11, the distance 
between isobars on a chart is measured in degrees of latitude (an average 
spacing over a fetch is ordinarily used), and the latitude position of the 
fetch is determined. Using the spacing as ordinate and location as abscissa, 
the plotted or interpolated slant line at the intersection of these two 
values gives the geostrophic wind speed. For example, in Figure 3-9, a 
chart with 3-millibar isobar spacing, the average isobar spacing (measured 
normal to the isobars) over F2, located at 37°N. latitude, is 0.70° of 
latitude. Using the scales on the bottom and right side of Figure 3-11, 
a geostrophic wind of 67 knots is found. 

Geostrophic wind speeds are generally higher than surface wind speeds. 
The following instructions, U.S. Fleet Weather Facility Manual (1966), are 
recommended for obtaining estimates of the surface wind speeds over the 
open sea from the geostrophic wind speeds: 

(a) For moderately curved to straight isobars - no correction is 
applied. 

(b) For great anticyclonic (clockwise movement about a high pressure 
center in Northern Hemisphere and counter-clockwise in Southern Hemisphere) 
curvature - add 10 percent to the geostrophic wind speed. 

(c) For great cyclonic (counter-clockwise movement about a low 
pressure center in Northern Hemisphere and clockwise in Southern Hemisphere) 
curvature - subtract 10 percent from the geostrophic wind speed. 

Frequently the curvature correction can be neglected since isobars 
over a fetch are often relatively straight. The gradient wind can always 
be computed if more refined computations are desired. 

To correct for air mass stability, the sea-air temperature difference 
must be computed. This can be done from ship reports in or near the fetch 
area, aided by climatic charts of average monthly sea surface temperatures 
when data are too scarce. The correction to be applied is given in Table 
3-1. (U.S. Fleet Weather Facility Manual, 1966.) 

Over oceans, the surface winds generally cross the isobars toward low 
pressure at an angle of 10° to 20°. 



3-24 



Wind speed ( 23 to 27 knots ) f f 



Cloud type (Dense cirrus 
Itches) 



True direction from whict) 

wind is blowing ^^ 

Current oir temperoture 

(31° F) jj 

Total onnount of cloud 

( Completely covered ) |^ 

Visibility in miles and 

froction (3/4 mile) VV-»-^^^ 

Present weottier. 

Continuous ligtit snow WW 

Temperature of dew 

point ( 30° F) T(J 

Cloud type ( Froctocumulus) C 



Height at base of cloud 

(2 = 300 to 599 feet) ^ 




'' sky covered I 
iddle cloud 
(6 = 7 or 8 tenths) 



NOTE : The letter symbols for eoch weather 
element ore shown above. 



Courtesy 
abridged 



Cloud type (Altostrotus) 



Barometric pressure in 
tenths of millibars reduced 
to sea level. Initial 9or 10 
and the decimal point ore 
ppp omitted (247= 1024.7 mb) 

Pressure change in 3 hours 
proceeding observotion 
_(28 : 2.8mb) 

Characteristic of barograph 
trace (Foiling or steody, 
then rising, or rising, then 
_rising more quickly) 

Plus or minus sign showing 

whether pressure is higher 

_or lower then 3 hours ago 

Time precipitation begon 
_or ended (4 = 3104 hrs.ogo) 



yu Post weather (Rain) 

Amount of precipitation 
pp (45= 0.45 inches) 



United Stotes Weather Bureau 
from W.M.O. Code 



Figure 3-10. Sample Plotted Report 



3-25 



For 



Ur 



T : lO^C 

Ap : 3 mb and 4mb 

An : isobar spacing measured in 



I Ap 

/'nf An 



where 



- angular velocity of earth, 
0.2625 rodians/hour 



degrees latitude 

1013.3 mb 

1. 247X10"' gm/cm' 

f : Coriolis parameter : 2 wsin 

0.3 



- latitude in degrees 



P 




30 35 40 45 50 55 60 70 

Degrees Latitude 

(offer Bretschneider, 1952a) 

Figure 3-11. Geostrophic Wind Scale 



3-26 



Table 3-1. Correction for Sea-Air Temperature^ 



Sea Temperature minus 
Air Temperature 


Ratio of Surface Wind Speed to 
Geostrophic Wind Speed, U/Ug 


or negative 

to 10 
10 to 20 
20 or above 


0.60 
0.65 
0.75 
0.90 



*Pore and Richardson (1969) and Hasse and Wagner (1971) 
report recent studies designed to refine the above table. 
Neither were able to find enough high quality observa- 
tions of large differences between air and sea tempera- 
tures to rigorously establish any effect of the sea-air 
temperature difference on the ratio of the surface wind 
speed to the gradient wind speed over the ocean. Both 
recommended the use of a constant value near 0.6 for 
most routine work. 

If there are several observed wind reports within the fetch region, 
and these consistently deviate in the same manner from wind speeds arrived 
at by the instructions given above, an average between the reported values 
and those computed by the above instructions will usually be the best 
estimate. 

Over the Great Lakes and some coastal regions, large temperature 
inversions (temperature increasing with elevation) may be observed. 
Bellaire (1965) reports air temperatures more than 15°C (27°F) greater 
than the water temperature in May 1964. When air temperature is much 
greater than that of the sea, all turbulent motion in the lower atmosphere 
is suppressed, and the wind near the surface has little relation to the 
wind estimate determined from a synoptic weather chart. Near mountainous 
coasts and particularly in fjords, the wind near the sea is often channeled 
to flow parallel to the mountains. The local temperature contrast between 
snow-covered mountains and relatively warm open water may have more control 
over the wind near the water than the isobaric pressure pattern from 
weather maps. In these cases, the wind determined from the pressure 
analysis on a weather map has little if any value for wave prediction. 
For these exceptional cases, there is no valid substitute for wind 
observations. 

3.42 DELINEATING A FETCH 

The fetch has been defined subjectively as a region in which the wind 
speed and direction are reasonably constant. Confidence in the computed 
results begins to deteriorate slightly when wind direction variations 
exceed 15°, and deteriorates significantly when direction deviations 
exceeding 45° are accepted in the fetch area. The computed results are 
sensitive to changes in wind speed as small as 1 knot (0.5 meter/second), 
but it is not possible to estimate the wind speed over any sizable region 



3-27 



with this precision. For practical wave predictions it is usually satis- 
factory to regard the wind speed as reasonably constant if variations do 
not exceed 5 knots (2.5 meters/second) from the mean. A coastline upwind 
from the point of interest always limits a fetch. An upwind limit to the 
fetch may also be provided by curvature or spreading of the isobars as 
indicated in Figure 3-12 (Shields and Burdwell, 1970), or by a definite 
shift in wind direction. Frequently the discontinuity at a weather front 
will limit a fetch, although this is not always so. 



1004 '9°° 996 




996 



1000 



1004 



1004 




1020 1016 



., 1012 1016 



1006 




1020 ' 1016 



1012 '006 




1012 



Figure 3-12. Possible Fetch Limitations 

Estimates of the duration of the wind are also needed for wave predic- 
tion. Computed results, especially for short durations and high wind speeds 
may be sensitive to differences of only a few minutes in the duration. Com- 
plete synoptic weather charts are prepared only at 6-hour intervals. Thus 
interpolation between charts to determine the duration may be necessary. 
Linear interpolation is adequate for most uses, and, when not obviously 
incorrect, is usually the best procedure. 



3-28 



3.43 FORECASTS FOR LAKES, BAYS, AND ESTUARIES 

3.431 Wind Data . The techniques referred to for determination of wind 
speeds and directions from isoharic patterns apply generally to ocean 
areas. The friction that causes winds to spiral when crossing isobars 
and to have a velocity lower than geostrophic or gradient winds is more 
variable over land areas. When a fetch is close to land, this variability 
will alter anticipated wind directions and velocities. In enclosed or 
semienclosed bodies of water, such as lakes and bays, wind speeds and 
directions should be taken from actual weather station reports whenever 
possible. 

In enclosed bodies of water, or in other areas where the wind blows 
off the land, differing frictional effects of land and water should be 
considered, and indicated wind speeds should be adjusted for these effects. 
Studies by Myers (1954) and Graham and Nunn (1959) indicate recommended 
adjustments in wind speeds. (See Table 3-2.) The adjustment factor may 
vary considerably depending on the shoreline frictional characteristics. 
This adjustment is used only for short fetches such as those in reservoirs 
and small lakes. 

Often, over small or well-defined fetch areas, it is not convenient 
or even possible to utilize surface charts to determine wind characteris- 
tics. Where wind records exist for locations in or near a fetch area, 
these may be utilized. The accuracy of the forecast will depend on the 
completeness of the records, the extent of fetch, and the wave prediction 
technique employed. Where wind duration records are not available, local 
wind speed reports may still be utilized to forecast waves assuming un- 
limited durations, that is, wave growth is limited by the available fetch. 
Wave characteristics deduced in this way are only qualitative. 



Table 3-2. 


Wind-Speed Adjustment, Nearshore 


Wind Direction 


Location of Wind Station 


Ratio* 


Onshore 


2 to 3 miles offshore 


1.0 


Onshore 


At coast 


0.9 


Onshore 


5 to 10 miles inland 


0.7 


Offshore 


At coast 


0.7 


Offshore 


10 miles offshore 


1.0 



(Graham and Nunn, 1959) 
*Ratio of wind speed at location to overwater wind speed 
(both at 30- ft. level) . 

3.432 Effective Fetch . The effect of fetch width or limiting ocean wave 
growth in a generating area may usually be neglected since nearly all ocean 
fetches have widths about as large as their lengths. In inland waters 
(bays, rivers, lakes, and reservoirs), fetches are limited by land forms 
surrounding the body of water. Fetches that are long in comparison to 
width are frequently found, and the fetch width may become quite important, 
resulting in wave generation significantly lower than that expected from 
the same generating conditions over more open waters. 



3-29 



Saville (1954) proposed a method to determine the effect of fetch 
width on wave generation. Figure 3-13, based on this method, indicates 
the effective fetch for a relatively uniform fetch width. The following 
problem demonstrates the use of Figure 3-13. 

************** EXAMPLE PROBLEM ************* 

GIVEN : Consider a channel with a fetch length F = 20 miles, a width 
W = 5 miles, an average depth d = 35 feet, and a windspeed U = 50 mph 
along the long axis. 

FIND : Estimate the significant wave height Hg, and the significant 
wave period Tg. 

SOLUTION : Compute W/F = 5/20 =0.25 

From Figure 3-13 for W/F = 0.25, F^/F =0.45 

Compute F^. = 0.45 X 20 = 9 miles or 47,500 feet. 

Using the forecasting relations given in Section 3.6, Wave Forecasting 
for Shallow Water, for a fetch of 47,500 feet and a wind speed of 
50 mph and an average uniform depth of 35 feet, the significant wave 
height may be determined from Figure 3-27 to be Hg = 5.2 feet, say 
5 feet and the significant wave period will be T = 4.6 seconds, say 
5 seconds . 



The preceding example presents a simplified method of determining the 
effective fetch. Shorelines are usually irregular, and the uniform-width 
method indicated in Figure 3-13 is not applicable. A more general method 
must be applied. This method is based on the concept that the width of a 
fetch in reservoirs normally places a very definite restriction on the 
length of the effective fetch; the less the width-length ratio, the shorter 
the effective fetch. A procedure for determining the effective fetch 
distance is illustrated in Figure 3-14. It consists of constructing 15 
radials from the wave station at intervals of 6° (limited by an angle of 
45° on either side of the wind direction) and extending these radials 
until they first intersect the shoreline. The component of length of 
each radial in a direction parallel to the wind direction is measured and 
multiplied by the cosine of the angle between the radial and the wind 
direction. The resulting values for each radial are summed and divided 
by the sum of the cosines of all the individual angles. This method is 
based on the following assumptions: 

(a) Wind moving over a water surface transfers energy to the 
water surface in the direction of the wind and in all directions within 
450 on either side of the wind direction. 

(b) The wind transfers a unit amount of energy to the water along 
the central radial in the direction of the wind and along any other radial 
an amount modified by the cosine of the angle between the radial and the 
wind direction. 

3-30 











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3-31 




( U.S. Army, B.E.B. Tech. Memo No 132,1962) 
Figure 3-14. Computation of Effective Fetch for Irregular Shoreline 



3-32 



(c) Waves are completely absorbed at shorelines. 

Fetch distances determined in this manner usually are less than those 
based on maximum straight- line distances over open water. This is true 
because the width of the fetch places restrictions on the total amount of 
energy transferred from wind to water until the fetch width exceeds twice 
the fetch length. 

While 6° spacing of the radials is used in this example, any other 
angular spacing could be used in the same procedure. 

3.5 SIMPLIFIED WAVE-PREDICTION MODELS 

Use of the wave-prediction models discussed in Section 3.3, Wave 
Field, requires an enormous computational effort and more meteorological 
data than one is likely to find outside of a major forecasting center. 
The Fleet Numerical Weather Center, Monterey, California began using this 
model on an experimental basis for a small part of the globe early in 1972. 
Expansion to larger regions is planned. Wave prediction begins with a com- 
putation of the existing wave field (often called a zero-time prediction), 
and continues with a calculation of the effects of predicted winds on the 
waves. A few years after this system is operational, it should be possible 
to supply the needs for wave-hindcast statistics by compilations of zero- 
time predictions. In the meantime, engineers who require wave statistics 
derived by hindcasting techniques for design consideration must accept 
simpler techniques. 

Computational effort required for the model discussed in Section 3.31, 
Development of a Wave Field, can be greatly reduced by the use of simpli- 
fied assumptions with only a slight loss in accuracy for wave height cal- 
culations, but sometimes with significant loss of detail on the distribu- 
tion of wave energy with frequency. One commonly used approach is to 
assume that both duration and fetch are large enough to permit an equi- 
librium state between the mean wind, turbulence, and waves. If this 
condition exists, all other variables are determined by the wind speed. 

Pierson and Moskowitz (1964) consider three analytic expressions which 
satisfy all of the theoretical constraints for an equilibrium spectrum. 
Empirical data, described by Moskowitz (1964) were used to show that the 
most satisfactory of these is 

-flCw^/w*) 
E(co)da; = (ag^/co*) e " dco , (3-20) 

where a and 3 are dimensionless constants, a = 8.1 x 10 3, g = 0.74 
and too - g/U, where g is the acceleration of gravity and U is the 
wind speed reported by weather ships, and o) is the wave frequency 
considered. 

Equation 3-20 may be expressed in many other forms. Bretschneider 
(1959, 1963) gave an equivalent form, but with different values for a 
and 3. A similar expression was also given by Roll and Fischer (1956). 
The condition in which waves are in equilibrium with the wind is called a 

3-33 



fully arisen sea. The assumption of a universal form for the fully arisen 
sea permits the computation of other wave characteristics such as total 
wave energy, significant wave height, and period of maximum energy. The 
equilibrium state between wind and waves rarely occurs in the ocean, and 
may never occur for higher wind speeds. 

A more general model may be constructed by assuming that the sea is 
calm when the wind begins to blow. Integration of the equations governing 
wave growth then permits the consideration of changes in the shape of the 
spectrum with increasing fetch and duration. If enough wave and wind 
records are available, empirical data may be analyzed to provide similar 
information. Pierson, Neumann, and James (1955) introduced this type of 
wave prediction scheme based almost entirely on empirical data. Inoue 
(1966, 1967) repeated this exercise in a manner more consistent with the 
Miles-Phillips theory using a differential equation for wave growth. Inoue 
was a member of Pierson 's group when this work was carried out, and his 
prediction scheme may be regarded as a replacement for the earlier 
Pierson-Neuman-James (PNJ) wave prediction model. The topic has been 
extended by Silvester and Vongvisessomjai (1971) and others. 

These simplified wave prediction schemes are based on the implicit 
assumption that the waves being considered are due entirely to a wind 
blowing at constant speed and direction for an overwater distance called 
the fetch and for a time period called the duration. 

In principle it would be possible to consider some effects of variable 
wind velocity hy tracing each wave train. Once waves leave a generating 
area and become swell, the wave energy is then propagated according to the 
group velocity. The total energy at a point, and the square of the signif- 
icant wave height could be obtained by adding contributions from individual 
wave trains. Without a computer, this procedure is too laborious, and 
theoretically inaccurate. 

A more practical procedure is to relax the restrictions implied by 
derivation of these schemes. Thus wind direction may be considered con- 
stant if it varies from the mean by less than some finite value, say 30°. 
Wind speed may be considered constant if it varies from the mean by less 
than ± 5 knots (2.5 meters/second) or h barb on the weather map. This 
assumption is not much greater than the uncertainty inherent in wind 
reports from ships. In this procedure, average values are used and are 
assumed constant over the fetch area, and for a particular duration. 

The theoretical spectra for the partially arisen sea can be used 
to develop formulas for such wave parameters as total energy, significant 
wave height and period of maximum energy density. 

Similar formulas can also be developed empirically from wind and wave 
observations. A quasi-empirical - quasi-theoretical procedure was used by 
Sverdrup and Munk (1947) to construct the first widely used wave predic- 
tion system. The Sverdrup-Munk prediction curves were revised by 
Bretschneider (1952a, 1958) with additional empirical data. Thus, this 
prediction system is often called the Sverdrup-Munk-Bretschneider (SMB) 
method. It is the most convenient wave prediction system to use when a 
limited amount of data and time are available. 



3-34 



3.51 SMB METHOD FOR PREDICTING WAVES IN DEEP WATER 

Revisions of earlier SMB forecasting curves are seen in Figures 3-15 
and 3-16. The curves represent dimensional plots from the empirical 
equations. 



^ 
U^ 



= 0.283 tanh 



0.0125 




0.42 



(3-21) 



2nU 



= 1.20 tanh 



0.077 



'gF 



,0.25 



(3-22) 



and. 



U 



= Kexp 




B In 




^ 



+ D b 






(3-23) 



where 



exp (x) = 



In 
K 
A 
B 
C 



loge 
6.5882 

0.0161 

0.3692 

2.2024 



and 



D = 0.8798 



With these relationships, the significant wave height H^, and significant 
wave period T^, at the end of a fetch may be estimated when wind speed, 
fetch length, and duration of wind over the fetch are given. Estimation 
of wind parameters is discussed in Section 3.4, Wind Information Needed 
for Wave Prediction, following the evaluation of simplified prediction 
techniques. In using Figures 3-15 or 3-16 the estimated wind velocity U, 
the fetch length F, and the estimated wind duration t, when a fetch 
first appears on a weather chart, are tabulated. Figure 3-15 or 3-16 is 
then entered with the value of U, using the scale on the left if U is 
in knots, or the scale on the right if U is in statute miles per hour. 
This U line is then followed from the left side of the graph across to 
its intersection with the fetch length or F line, or the duration t 
line, whichever comes first. 



3-35 



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3-37 



************** EXAMPLE PROBLEM ************* 

GIVEN : A wind speed U = 35 knots (40 mph) and a duration t = 10 hours . 

FIND : The significant wave height H^, and the significant period T^^, 
at the fetch front for: 

(a) A fetch length, F = 200 nautical miles and 

(b) a fetch length, F = 80 nautical miles. 
SOLUTION : 

(a) Enter Figure 3-15 from the left side at U = 35 knots and move 
horizontally across the figure from the left toward the right, until 
intersecting the dashed line for a duration of 10 hours that comes 
before the line indicating a fetch length of 200 nautical miles. At 
the 10-hour duration line F = 92 nautical miles; this is the minimum 
fetch F^ for this case. 

With U = 35 knots, t = 10 hours, and F = 200 nautical miles then 
H^ = 13.1 feet, T^ = 8.0 seconds, t^ equals 10 hours, and F^ = 92 
nautical miles. 

(b) Entering Figure 3-15 as above, when F = 80 nautical miles 
and t = 10 hours, then the heights, periods, minimum duration and 
fetch would be H^ = 12.6 feet, T^ = 7.8 seconds, and t^ = 9.0 hours. 
The minimum duration t^ is 9 hours, corresponding to the miles 
which limit generation, and comes before a duration of 10 hours. 

In this example problem, the wave pattern in (a) is limited by the 
duration; the wave pattern in (b) is limited by the fetch. 



When a series of surface synoptic weather charts (Fig. 3-9) are used 
to determine wave patterns, the values of U, F and t can be tabulated 
for the first chart. For the same fetch on a later chart drawn for a time 
Z, after the first chart, U, F, and t are again tabulated. Using 
the subscript 2 to refer to those of the second chart and subscript 1 
to refer to those of the first chart, if U2 = U-^ , the above procedures 
should be followed using either t2 = t^i + Z or F2. If, however, U2 f U^ , 
certain additional assumptions must be made before using the forecasting 
curves . 

A change in wind speed from Uj^ to U2 in a time Z between charts may 
be assumed to take place instantaneously at a time Z/2. Waves due to Ui 
may then be calculated by assuming that the first chart minimum duration 
time has been lengthened by an amount Z/2 or that its minimum fetch has 
been changed by AF/2, where AF represents the change in fetch length 
between weather charts. Since at the assumed abrupt change in wind speed. 



3-38 



the energy imparted to the waves by U^ , with a minimiim duration t^^ + Z/2 
for a minimum fetch F^^ + F/2, does not change, U2 will be assumed to 
impart energy to waves which already contain energy due to Uj^. 

Plotted on Figures 3-15 and 3-16 are dotted lines of constant H^T'^ which 
are considered lines of constant wave energy. To a first approximation, 

deepwater wave energy is given by 

E - ^^ = M2pg(m: . 
"88 

If energy had been imparted to the waves by U2 acting alone, these waves 
would be of length and height given in Figures 3-15 or 3-16 by the inter- 
section of the U2 ordinate with the constant energy line (plotted or 
interpolated) corresponding to energy imparted by Ui with a minimum 
duration of t^-^ + Z/2 or a minimum fetch F^^ + F/2. By increasing 
the minimum duration at this point by an amount Z/2 or by changing the 
minimum fetch by an amount AF/2, wave conditions under U2 at the 
time of the second chart may be approximated. 

For example, if the wind speed increases so that U2 = 40 knots, and 
with Ui = 35 knots, t^-^ = 10 hours, F^^ = 92 nautical miles, t^^ + Z/2 = 
13 hours; an interpolated (by eye) dotted line of constant H^T^ would be 
followed up to the LI2 = 40-knot line where the duration = 6.5 hours. To 
this value Z/2 or 3 hours is added and then moving horizontally along 
the line U2 = 40 knots to t = 6.5 + 3.0 = 9.5 hours, it is found that 

H^2 - 15.6 feet, T^2 ~ ^-^ seconds, t^2 ~ ^-^ hours, and F^2 - ^^ nautical 
miles. If the measured fetch F2 had been less than 95 nautical miles, 
this length of fetch would limit the growth of waves. Although the pre- 
ceding discussion would indicate that AF should be calculated, in practice 
this need not be done; the results obtained through calculation of AF 
would be found by reading off wave heights at the intersection of U2 and 
F2 if F2 is limiting. Therefore, if F2 had equalled 85 nautical miles, 
in this case less than 95 miles and therefore limiting, at the intersection 
of U2 = 40 knots, then Hp2 = 15.0 feet, T^2 =8.5 seconds, T^2 = 8. 8 hours, 
and F^2 - 85 nautical miles. Note this important distinction: t;^, F^ and Fj 
are calculated by use of Figure 3-15. Some of the measured and calculated 
values will be the same, but not all of them. 

If the wind velocity U2 is less than Ui , the procedures followed 
are nearly the same. From the intersection of U-^ and t^-^ + Z/2 a 
constant energy line is followed to its intersection, if there is one. 



3-39 



with either U2 or F2 whichever comes first from the left side of the 
figure. If U comes first, Z/2 is added to the duration at this point, 
and the U2 ordinate is followed to either this new duration or to the 
F2 whichever is first from the left side of the chart. (Compare with the 
preceding paragraph.) At this point, Hp2' ^F2' hn2> ^^^ ^m2 inclusive, 
are read off. If the constant energy line had intersected F2 before U2, 
it is only necessary to drop down along the F2 abscissa to its intersec- 
tion ^'ith U2, and at this point read H^2> '^F2> ^2 ^^^ ^m2- (This pro- 
cedure could be used for many cases in which U2 is greater than Uj.) 

The major differences in technique are used when U2 is less than Ui 
and the H^T^ = constant line from the intersection of Ui and tmi + Z/2 
does not intersect either U^ or F . Forecasting theory used here pre- 
dicts that waves due to a constant wind blowing over an unlimited fetch 
for an unlimited duration will eventually reach limiting height and period 
distributions beyond which growth will not continue. In Figure 3-16 the 
limit of this state is delineated by the line labelled maximum condition. 
To the right of this line, it is assumed that any energy transport to the 
waves by the wind is compensated by wave breaking, hence no wave growth 
occurs . 

3.52 EFFECTS OF MOVING STORMS AND A VARIABLE WIND SPEED AND DIRECTION 

In principle, it should be possible to extend the Inoue differential 
equation for wave growth to highly irregular conditions, but no experi- 
mental verification of this concept has been published. Kaplan (1953) 
and Wilson (1955) have proposed techniques for applying the simplified 
prediction techniques to variable wind fields and changing fetches. The 
procedures appear reasonable and these techniques are used, although no 
statistics are available for verification. 

3.53 VERIFICATION OF SIMPLIFIED WAVE HINDCAST PROCEDURES 

Comparisons of hindcast wave heights and observed wave heights, 
similar to Figure 3-7, have been given by Jacobs (1965) for the PNJ wave- 
prediction system, by Bates (1949) and Isaacs and Saville (1949) for the 
early Sverdrup-Munk method, by Kaplan and Saville (1954), for the early 
SMB method, and by Bretschneider (1965) for a later revision. The basic 
data from which the prediction curves were derived, siommarized by 
Bretschneider (1951), also indicate the range of variation that may be 
expected. 

It is generally believed that much of the discrepancy between observed 
and predicted waves is random, and that statistical summaries of observa- 
tions and predicted values will agree much better than the individual 
observations. Saville (1954) and Pierson, Neumann and James (1955) give 
summaries of results from a systematic program for deepwater hindcasting 
waves at the four locations shown in Figure 3-17. The U.S. Naval Weather 



3-40 




STATION A^ 

oo; 




:CAPE COD, 



STATION B 

4I°50' 
69° 30' 



■42= 



66' 



AREA 4 



.NEW YORK/ 



40" 



STATION C| 

40° 15' 
73°45* 



72" 



69" 

"AREA 5 



AREA 6 




-AREA 8 



^ 



.V 



A 



\ 



A 



V 



Figure 3-17. Location of Wave Hindcasting Stations and Summary of 
Synoptic Meterological Observations (SSMO] Areas 



3-4 



Service Command (1970) provides summaries of shipboard wave observations. 
Summary of Synoptic Meteorological Observations (SSMO) for the hatched 
areas indicated in Figure 3-17. Cumulative distribution functions for 
wave heights as determined by both hindcasting techniques and the ship- 
board observations are given in Figure 3-18. The average of the two 
forecasting methods agrees reasonably well with the shipboard observations. 

3.54 ESTIMATING WAVE DECAY IN DEEP WATER 

Figures 3-19 and 3-20 are used to estimate wave characteristics after 

the waves have left the fetch area but are still travelling in deep water. 

With Figure 3-19, and given H^, T^, F^ and D (the decay distance), it is 
possible to compute the ratios 

decayed wave height D decayed wave period D 

—z — = — , and — -: — — = — 

fetch wave height Hj, fetch wave period Tp 

With Figure 3-20, it is possible to compute wave travel time between 
a fetch and a coast, knowing the decayed wave period T^ and the decay 
distance D. 

This tvavet time t^ is determined by dividing the decay distance 
by the deepwater group velocity for waves having a period equal to the 
decayed period Ij^. These values enable the estimation of arrival times 
for waves at the end of the decay distance. 

Waves, after leaving a generating area, will generally follow a great- 
circle path toward a coast. However, sufficient accuracy is usually 
obtained by assuming wave travel in a straight line on the synoptic chart. 
Decay distance is found by measuring the straight line distance between 
the front of a fetch and the point for which the forecast is being made. 
If a forecast is being made for a coastal area, the effects of shoaling, 
refraction, bottom friction and percolation will have to be considered in 
translating the deepwater forecast to the shore. 

3.6 WAVE FORECASTING FOR SHALLOW WATER 

3.61 FORECASTING CURVES 

Water depth affects wave generation. For a given set of wind and 
fetch conditions, wave heights will be smaller and wave periods shorter if 
generation takes place in transitional or shallow water rather than in deep 
water. Several forecasting approaches have been made; the method given by 
Bretschneider as modified using the results of Ijima and Tang (1966) is 
presented here. Bretschneider and Reid (1953) consider bottom friction 
and percolation in the permeable sea bottom. 

There is no single theoretical development for determining the actual 
growth of waves generated by winds blowing over relatively shallow water. 
The numerical method presented here is based on successive approximations 



3-42 



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22 



3-45 



in which wave energy is added due to wind stress and subtracted due to 
bottom friction and percolation. This method uses deepwater forecasting 
relationships originally developed by Sverdrup and Munk (1947) and revised 
by Bretschneider (1951) to determine the energy added due to wind stress. 
Wave energy lost due to bottom friction and percolation is determined from 
the relationships developed by Bretschneider and Reid (1953). Resultant 
wave heights and periods are obtained by combining the above relationships 
by numerical methods. The basic assumptions applicable to development of 
deepwater wave-generation relationships (Bretschneider, 1952b) as well as 
development of relationships for bottom friction loss (Putnam and Johnson, 
1949) and percolation loss (Putnam, 1949) apply. 



The choice of an appropriate bottom friction factor f^ for use in 
the forecasting technique is a matter of judgement; a value of f^ = 0.01 
has been used for the preparation of Figures 3-21 through 3-30 which are 
forecasting curves for shallow-water areas of constant depth. These curves, 
which may be used like Figures 3-15 and 3-16, are given by the equations: 



and 



^ 
U^ 



= 0.283 tanh 



0.530 1^^ 



2-nU 



= 1.20 tanh 



0.833 



,0.75 



,U^ 



tanh 



gd \ 0.375 



tanh 




(3-25) 



■ ,(3-26) 



which in deep water reduce to Equations 3-21 and 3-22 respectively. 



************** 



EXAMPLE PROBLEM ************** 



GIVEN : Fetch, F = 80,000 feet, wind speed, U = 50 mph . , water depth, 
d = 35 feet (average constant depth), bottom friction factor f^ = 0.01 
(assumed) . 

FIND : Wave height H and wave period T. 



3-46 



-H = 2.5ft. 



■Tr 4 25 sec 



1 — I — ! I^-L M 



^1 - ^ ^■o sec 




15 



2 25 3 4 

X 1,000 



5 6 7 8 9 10 1.5 

Fetch (F) feet 



2 25 3 4 

X 10,000 



5 6 7 8 9 10 



Figure 3-21. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 5 feet. 



T-5 25sec^ 
H;4.5ft.~.^ T = 5.05ec^ / 




1.5 2 2.5 3 4 5 6 7 8 9 10 1.5 2 2.5 3 4 5 6 7 8910 

XI, 000 Fetch (F) feet X 10,000 



Figure 3-22. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 10 feet. 



3-47 



H = 70ft 
1 = 6 sec. 




2 2.5 3 4 

X 1,000 



5 6 7 8 9 10 1.5 

Fetch (F) feet 



2 2.5 3 4 

X 10,000 



5 6 7 8 9 10 



Figure 3-23. Forecasting Cr.rves for Shallow-Water Wave-; 
Constant Depth = 15 feet. 



H = 5 75 ft. 




1.5 2 2.5 3 4 5 6 7 8 9 10 1.5 2 2.5 3 4 5 6 7 8 9 10 

X 1,000 Fetch (F) feet X 10,000 



Figure 3-24. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 20 feet. 



3-48 



/T=4 5sec. T 
= 4.0 secV 1 = 5 setv.,^^ 



= 5.5 sec.\ 



T = 6.0 sec.^ ,H = 8.0 ft. 

T = 6.5 sec.' 




1.5 



2 2.5 3 4 

X 1,000 



5 6 7 8 9 10 15 

Fetch (F) feet 



2 2.5 3 4 5 6 7 8 9 10 

X 10,000 



Figure 3-25. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 25 feet. 



H-8.0 ft.> 




1.5 2 2.5 3 4 5 6 7 8 9 10 1.5 2 2.5 3 4 5 6 7 8 9 10 

X 1,000 Fetch (F) feet X 10,000 



Figure 3-26. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 30 feet. 



3-49 



,H-90 ft. 




H = IOft 

= 7 sec 



15 2 2 5 3 4 5 6 7 8 9 10 15 2 2 5 3 4 

X 1,000 Fetch (F) feet X 10,000 



5 6 7 8 9 10 



l-'igure 3-27. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 35 feet. 



H-IOOft-, H-II.Oft. 
T = 6-5 sec.^ fT= 7 1 




l'i;iure 3-28. Forecasting Curves for Shallow-Water Waves 
Constnnt Pcpth = -10 feet. 



3-50 



T-7.0 secs 
T = 6 sec. H = ILO ft.N \ 
't:6 5 sec 




Figure 3-29. Forecasting Curves for Shallow-Water Waves 
Constant Depth = 45 feet. 




figure 3-30. 



Forecasting Curves for Shallow-Water Waves 
Constant Depth = 50 feet. 



3-51 



SOLUTION : From Figure 3-27 for constant depth, d = 35 feet, 

for 

F = 80,000 feet, 
and 

U = 50 mph. 
Then 

H = 5.9 feet, say 6 feet, 
and 

T = 5.1 seconds, say 5 seconds. 

***************************** ***k**f* 

3.62 DECAY IN LAKES, BAYS, AND ESTUARIES 

Section 3.33. Decay of A Wave Field applies to water aroa^^ contijiuous 
with land as well as those in the open ocean. Most fetches in inltu'.d 
waters will be limited at the front and at the rear by ^ l.ind m;i<s .uul 
decay distances will usually be relatively small or nonexistent. 

3.7 HURRICANE WAVES 

When predicting wave generation by hurricanes, the determination of 
fetch and duration from a wind field is more difficult than fur more normal 
weather conditions discussed earlier. The largo changes in wind speed and 
direction with both location and time cause the difficulty. Estimation of 
the free air wind field must be approached through matliematical models, 
because of the scarcity of observations in severe storms. However, the 
vertical temperature profile and atmospheric turbulence characte;-istics 
associated with hurricanes differ less from one storm to another than for 
other types of storms. Thus the relation between the free air winds and 
the surface winds is less variable for hurricanes than for other storms. 

3.71 DESCRIPTION OF HURRICANE WAVES 

In hurricanes, fetch areas in which wind speed and direction remain 
reasonably constant are always small; a fully arisen sea state never 
develops. In the high-wind zones of a storm, however, long-period waves 
which can outrun the storm may be developed within fetches of 10 to 20 
miles and over durations of 1 to 2 hours. The wave field in front, or to 
either side, of the storm center will consist of a locally generated sea, 
and a swell from other regions of the storm. Samples of wave spectra, 
obtained during hurricane Agnes, 1972, are shown in Figure 3-31. Most 
of the spectra display evidence of two or three distinct wave trains; thus, 
the physical implications of a signifiaant wave period is not clear. 

Other hurricane wave spectra computed with an analog spectrum analyzer 
from wave records obtained during Hurricane Donna, 1959. have been published 
by Bretschneider (1963) . Most of these spectra also contained two distinct 
peaks . 



3-52 



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0.2 



0.4 



Figure 3-31. Typical Hurricane Wave Spectra. Typical Hurricane 
Wave Spectra from the Atlantic Coast of the United 
States. The ordinate scale is the fraction of 
total wave energy in each frequency band of 0.0011 
Hertz (one Hertz is one cycle/second). A linear 
frequency scale is shown at bottom of each graph 
and a non-linear period scale at top of each graph. 



3-53 



An indication of the distribution of waves throughout a hurricane can 
be obtained by plotting composite charts of shipboard wave observations. 
The position of a report is determined by its distance from the storm 
center and its direction from the storm track. Changes in storm intensity 
and shape are often small enough to permit all observations obtained during 
a period of 24 to 36 hours to be plotted on a single chart. Several plots 
of this type from Pore (1957) are given in Figure 3-32. Additional data 
of the same type have been presented by Arakawa and Suda (1953) , Pore (1957) 
and Harris (1962) . 

Goodknight and Russell (1963) give a tabulation of the significant 
height and period for waves recorded on an oil drilling platform in approxi- 
mately 33 feet of water, 1.5 miles from shore near Burrwood, Louisiana 
during hurricanes Audrey, 1957, and Ella, 1950, and tropical storms Bertha, 
1957, and Esther, 1957. These wave records were used to evaluate the 
applicability of the Rayleigh distribution function (Section 3.22. Wave 
Height Variability) to hurricane statistics for wave heights and periods. 
They concluded that the Rayleigh distribution function is adequate for 
deriving the ratios between Hg, Hio, H, etc., with sufficient accuracy 
for engineering design, but that its acceptance as a basic law for wave 
height distributions is questionable. 

3.72 MODEL WIND AND PRESSURE FIELDS FOR HURRICANES 

Many mathematical models have been proposed, for use in studying 
hurricanes. Each is designed to simulate some aspect of the storm as 
accurately as possible without making excessively large errors in describ- 
ing other aspects of the storm. Each model leads to a slightly different 
specification of the surface wind field. Available wind data are suffi- 
cient to show that some models duplicate certain aspects of the wind field 
better than certain other models; but there are not enough data for a 
determination of a best model for all purposes. 

One of the simplest and earliest models for the hurricane wind field 
is the Rankin vortex. For this model, it is assumed that 

U = Kr for r < R , 

KR^ (3-27) 

U = for r > R , 

r 

where K is a constant, R is the radial distance from the storm center 
to the region of maximum wind speed, and r is the radial distance from 
the storm center to any specified point in the storm system. 

This model can be improved by adding a translational component to 
account for storm movement and a term producing cross-isobar flow toward 
the storm center. 

Extensions of this model are still being used in some engineering 
studies (Collins and Viehman, 1971). This model gives an artificial 
discontinuity in the horizontal gradient of the wind speed at the radius 
of maximum winds, and does not reproduce the well-known area of calm 
winds near the storm center. 

3-54 



STORM DIR 



,.V' 



_!^'>- '5/^/1. JL'"^- C CONNIE 1955 






AUG 10 0130E THBU v 
AUG 11 0130E 

PERIOD SCAIE IN SEC 
10 20 



?'^V 



Y 



r 



/• 



'A 



'h 

+-H— I — (tH — I— t^'- 



\ 



>K° 






/, \" '• 



STORM DIR '\ 



•■ X 



a CONNIE 1955 

AUG n 0730E THRU ..', 
AUG 12 0730E 



PERIOD SCALE IN SEC -- ..'> 



;^ 



-^^^ 



X'A. 



■•<:v 






.V 



\' 



>. s 



\10 \'» 



"'^v^:;vi;..^ A 



'•-A.^,;\. 



v'f^. 



.^' 



■ivii: ^'' '»i 1 I 



, i to n 

i3' 



■rtf; 
K' . ... 



•r. 



STORM OIR 






\ 



,./ 



'l •^. ' "^ AUG 15 1330E THRU 



-P^^MtT 



e DIANE 1955 .l>"/rt 

..._ .. -- V» A 

A. . AUG 16 0730E af A 

PERIOD SCAIE IN SEC -S" '" ], \ 

fs 



10 20 



•y 



-rh 



„ ".'Z 

"-• Ml-" 



Figure 3-32. 



I I I I .,1 I % I 



STORM DIR 



f DIANE 1955 

AUG16. 1330E THRU 
AUG 17 0730E 



PERIOD SCAIE IN SEC 
10 20 



1 I L 



4» *30 MO 



->Ai 



WO IBO laj 60 



1. /. 

y a, 10, 



/■ 



y 



H — \ 









/^v\* ^ -vx T.;v v^ 
il>4: "i/ +"/ • '^ 



'/ 



:\'^ 



(offer Pore, 1957) 

Compositive Wave Charts. The wave height in feet is plotted 
beside the arrow indicating direction from which the waves 
came. The length of the arrow is proportional to the wave 
period. Dashed arrow indicates unknown period. Distances 
are marked along the radii at intervals of 60 nautical miles. 



3-55 



A more widely used model was given by Myers (1954). A concise mathe- 
matical description of this model is given by Harris (1958) as follows: 



(3-28) 





P - 


- P 




_ 


R 






o 






r 








^ 


c 


) 




P - 


- P 










n 


o 








U^ 








1 

Pa 




r 


- + 


*".. 


= 


('. 



'o)? 



R -^ 



e , (3-29) 



where p is the pressure at a point located at a distance r from the 
storm center, p is the central pressure, p is the pressure at the 
outskirts of the storm, p^ is the density of air, and U„p is the 

gradient wind speed. Agreement between this model and the characteristics 
of a well-observed hurricane is shown in Figure 3-33. The insert map gives 
the storm track; dots indicate the observed pressure at several stations in 
the vicinity of Lake Okeechobee, Florida; the solid line (Figi 3-33a) gives 
the theoretical pressure profile fitted to three points within 50 miles of 
the storm center. The corresponding theoretical wind profile is given by 
the upper curve of Figure 3- 33b. Observed winds at one station are indi- 
cated by dots below this curve. A solid line has been drawn through these 
dots by eye to obtain a smooth profile. The observed wind speed varies in 
a systematic way from about 65 percent of the computed wind speed at the 
outer edge to almost 90 percent of the predicted value near the zone of 
maximum wind speed. Reasonably good agreement between the theoretical and 
observed wind speeds has been obtained in only a few storms. This lack of 
agreement between the theoretical and observed winds is due in part to the 
elementary nature of the model, but perhaps equally to the lack of accurate 
wind records near the center of hurricanes. 

Parameters obtained from fitting this model to a large number of storms 
were given by Myers (1954). Parameters for these other storms (and for 
additional storms) are given by Harris (1958). Equation 3-29 will require 
some form of correction for a moving storm. 

This model is purely empirical, but it has been used extensively and 
it provides reasonable agreement with observations in many storms. Other 
equally valid models could be derived; however, alternative models should 
not be adopted without extensive testing. 

In the northern hemisphere, wind speeds to the right of the storm 
track are always higher than those on the left, and a correction is needed 
when any stationary storm model is being used for a moving storm. The 
effect of storm motion on the wind field decreases with distance from the 
zone of highest wind speeds. Thus the vectorial addition of storm motion 
to the wind field of the stationary storm is not satisfactory. Jelesnianski 
(1966) suggests the following simple form for this correction, 

^SM « = :^rh ^F . C3-30) 

rv. T r 

3-56 



ZZ^^e-"/^ 




1,2 

'0 



10 20 30 40 50 60 70 10 20 30 40 SO (O 

Distance From Pressure Center (Statute Miles) DIatonce From Wind Center (Statute Miles) 

Hurricone August 26-27,1949 ^ ( from Horri,, 1958) 

Figure 3-33. Pressure and Wind Distribution in Model Hurricane. Plotted 
dots represents observations 

where V„ is the velocity of the storm center, and Ugw(r) is the 
convective term which is to be added vectorially to the wind velocity at 
each value of r. Wilson (1955, 1961) and Bretschneider (1959, 1972) have 
suggested other correction terms. 



3.73 PREDICTION TECHNIQUE 



For a slowly moving hurricane, the following formulas can be used to 
obtain an estimate of the deepwater significant wave height and period at 
the point of maximum wind: 

RAp 



H = 16.5 e 
o 



100 



0.208 a Vc 



1 + 



Vu, 



(3-31) 



and 



T, = 8.6 e 
s 



RAp 
200 



1 + 



0.104 a Vp 



vuT 



(3-32) 



3-57 



where 

H^ = deepwater significant wave height in feet 

Tg = the corresponding significant wave period in seconds 

R = radius of maximum wind in nautical miles 

Ap = p - p , where p„ is the normal pressure of 29.92 inches 

of mercury, and p is the central pressure of the hurricane 

V„ = The forward speed of the hurricane in knots 

U^ = The maximum sustained wind speed in knots, calculated for 
30 feet above the mean sea surface at radius R where 

U^ = 0.865 U^^^ (For stationary hurricane) (3-33) 

U^ = 0.865 U^^ + 0.5 V^ (For moving hurricane) (3-34) 

^rmx ~ Maximum gradient wind speed in knots 30 feet above the 
water surface 

^max = 0.868 [73 (p„ - p^) 1/2 . R(0.575f)] (3-35) 

f = Coriolis parameter = 2to sin(j), where w = angular velocity of 
earth = 2ti/24 radians per hour 

Latitude (()>) 25° 30° 35° 40° 
f (rad/hr) 0.221 0.262 0.300 0.337 

a = a coefficient depending on the forward speed of the 
hurricane and the increase in effective fetch length, 
because the hurricane is moving. It is suggested that 
for slowly moving hurricane a = 1.0. 

Once Hq is determined for the point of maximum wind from Equation 
3-31 it is possible to obtain the approximate deepwater significant wave 
height H^ for other areas of the hurricane by use of Figure 3-34. 

The corresponding approximate wave period may be obtained from 

T =2.13 y/H^ (in seconds) , (3-36) 

where H^ is in feet (derived from empirical data showing that the wave 
steepness H/T^ will be about 0.22). 



3-58 




-M to 



Figure 3-34. Isolines of Relative Significant Wave 
Height for Slow Moving Hurricane 



3-59 



************** 



EXAMPLE PROBLEM ************** 



GIVEN : Consider a hurricane at latitude 350N. with R = 36 nautical miles, 
Ap = 29.92 - 27.61 = 2.31 inches of mercury, and V^, forward speed, = 
26 knots. Assume for simplicity that a = 1.0. 

FIND : The deepwater significant wave height and period. 
SOLUTION : 

Using Equation 3-35 



U 



U 



max 



0.868 [73/p -p Y' - R(0.575f)l 
0.868 [73(2.31)^^ - 36(0.575X0.300)1 



^max =" 0-^^8 (110.95 - 6.23) = 90.9 knots. 



Using Equation 3-34 



Ur = 0.865 U^,^ + 0.5 V^ 

Uj^ = 0.865 (90.9) + 0.5 (26) = 91.6 knots. 



Using Equation 3-31 



H^ = 16.5 e 



RAp 
100 



0.208 aVc 



1 + 



Vu 



R 



where the exponent 

RAp _ 36(2.31) 



100 



100 



= 0.832, 



and 



then 



eO.832 = 2.30 



H^ = 16.5 eO-832 



1 + 



0.208 X 1 X 26 



V9I.6 



Hq = 16.5 (2.30) (1.564) = 59.4 feet. 



3-60 



Using Equation 3-32 



T = 8.6 e 



RAp 
200 



1 + 



0.104 a Vjp 

Vu7 



where the exponent 

RAP 36(2.31) 



200 



200 



= 0.416 



T = 8.6 e 
s 



0.416 



1 + 



0.104 X 1 X 26 
V91.6' 



Tj = 8.6 (1.52) (1.282) 



16.8 seconds. 



Alternately, by Equation 3-36, it is seen that 



T = 2.I3V59.4 = 16.4 seconds. 

It should be noted that computing the values of wave height and period 
to three significant figures does not imply the degree of accuracy of the 
method; it is done to reduce the computational error. 



Referring to Figure 3-34, H^ = 59.4 feet corresponds to the relative 
significant wave height of 1.0 at r/R = 1.0, the point of maximum winds 
located, for this example, 36 nautical miles to the right of the hurricane 
center. At that point the wave height is about 60 feet, and the wave 
period T is about 16 seconds. At r/R = 1.0 to the left of the hurricane 
center, from Figure 3-34 the ratio of relative significant height is about 
0.62, whence H^ = 0.62 (59.4) = 36.8 feet. This wave is moving in a direc- 
tion opposite to that of the 59.4-foot wave . The significant wave period 
for the 36.8-foot wave is: Tg = 2.13 /36.8' = 12.9 seconds, say 13 seconds. 

The most probable maximum wave is assumed to depend on the number of 
waves considered applicable to the significant wave, H^ = 59.4 feet. This 
number N depends on the length of the section of the hurricane for which 
near steady state exists and the forward speed of the hurricane. It has 
been found that maximum wave conditions occur over a distance equal to the 
radius of maximum wind. The time it takes the radius of maximum wind to 
pass a particular point is 



t = 



36 

26 



= 1.38 hours = 4,970 seconds; 



(3-37) 



the number of waves will be 



t 4970 

N = — = 

T, 16.4 



303. 



(3-38) 



3-61 



The most probable maximum waves can be obtained by using 



Hn = 0-707 H^ ^ log^ - . 



(3-39) 



The most probable maximum wave is obtained by setting n = 1, and 
using Equation 3-39 



Hj = 0.707(59.4) 



j 30? 



100.4 feet, say 100 feet. 



Assuming that the 100-foot wave occurred, then the most probable 
second highest wave is obtained by setting n = 2, the third from n = 3, etc. 



Hj = 0.707(59.4) 



I 30F 
•Jlogg — — = 94.1 feet, say 94 feet; 



H3 = 0.707(59.4) ./log 



i 



303 



-3 , v-,..^ \l^^i>e -^ = 90.2 feet, say 90 feet. 



The problem now is to determine the changes in the deepwater waves as 
they cross the Continental Shelf, taking into account the combined effects 
of bottom friction, refraction, the continued action of the wind, and the 
forward speed of the hurricane. This requires numerical integration, using 
Table 3-3, Figure 3-35, and refraction diagrams. It is also necessary to 
obtain an effective fetch length, by use of 



Fe = 



H, 



0.0555 U, 



(3-40) 



where 



and 



F is the effective fetch in nautical miles, 

H^ is the deepwater significant wave height in feet, 



Uj^ is the maximum sustained wind speed in knots. 
For this example, using Equation 3-40 

59.4 



Fe = 



0.0555(91.6) 



= (11.69) = 137 nautical miles. 



3-62 









Table 3-3. 


Values of K or 


(ivh;) 








■^v 


0.00 


0.01 


0.02 


0.03 


0.04 


0.05 


0.06 


0.07 


0.08 


0.09 


0.1 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


0.2 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


1.000 


0.999 


0.999 


0.999 


0.3 


0.998 


0.998 


0.997 


0.997 


0.996 


0.995 


0.994 


0.993 


0.992 


0.991 


0.4 


0.990 


0.989 


0.987 


0.986 


0.984 


0.982 


0.981 


0.980 


0.978 


0.976 


0.5 


0.974 


0.973 


0.971 


0.969 


0.967 


0.965 


0.963 


0.961 


0.960 


0.959 


0.6 


0.957 


0.955 


0.954 


0.952 


0.951 


0.949 


0.948 


0.946 


0.945 


0.943 


0.7 


0.942 


0.941 


0.939 


0.938 


0.937 


0.936 


0.935 


0.934 


0.932 


0.931 


0.8 


0.930 


0.929 


0.929 


0.928 


0.927 


0.926 


0.925 


0.924 


0.924 


0.923 


0.9 


0.922 


0.922 


0.921 


0.921 


0.920 


0.919 


0.919 


0.918 


0.918 


0.917 


1.0 


0.917 


0.917 


0.916 


0.916 


0.916 


0.915 


0.915 


0.915 


0.915 


0.914 


1.1 


0.914 


0.914 


0.914 


0.914 


0.914 


0.913 


0.913 


0.913 


0.913 


0.913 


1.2 


0.913 


0.913 


0.913 


0.913 


0.913 


0.913 


0.913 


0.913 


0.913 


0.913 


1.3 


0.913 


0.913 


0.914 


0.914 


0.914 


0.914 


0.914 


0.914 


0.914 


0.915 


1.4 


0.915 


0.915 


0.915 


0.915 


0.915 


0.916 


0.916 


0.916 


0.916 


0.917 


1.5 


0.917 


0.917 


0.917 


0.918 


0.918 


0.918 


0.919 


0.919 


0.919 


0.919 


1.6 


0.920 


0.920 


0.920 


0.920 


0.921 


0.921 


0.921 


0.922 


0.922 


0.922 


1.7 


0.923 


0.923 


0.924 


0.924 


0.924 


0.925 


0.925 


0.925 


0.926 


0.926 


1.8 


0.927 


0.927 


0.927 


0.928 


0.928 


0.928 


0.929 


0.929 


0.930 


0.930 


1.9 


0.931 


0.931 


0.931 


0.932 


0.932 


0.933 


0.933 


0.933 


0.934 


0.934 


2.0 


0.935 


0.935 


0.936 


0.936 


0.936 


0.937 


0.937 


0.938 


0.938 


0.939 


2.1 


0.939 


0.940 


0.940 


0.941 


0.941 


0.942 


0.942 


0.942 


0.943 


0.943 


2.2 


0.943 


0.944 


0.945 


0.945 


0.946 


0.946 


0.947 


0.947 


0.947 


0.948 


2.3 


0.948 


0.949 


0.949 


0.950 


0.950 


0.951 


0.951 


0.951 


0.952 


0.952 


2.4 


0.953 


0.953 


0.954 


0.954 


0.955 


0.955 


0.956 


0.956 


0.956 


0.957 


2.5 


0.957 


0.958 


0.958 


0.959 


0.959 


0.960 


0.960 


0.961 


0.961 


0.962 


2.6 


0.962 


0.963 


0.963 


0.964 


0.964 


0.965 


0.965 


0.966 


0.966 


0.967 


2.7 


0.967 


0.968 


0.968 


0.969 


0.969 


0.969 


0.970 


0.970 


0.971 


0.971 


2.8 


0.972 


0.972 


0.973 


0.973 


0.974 


0.974 


0.975 


0.975 


0.976 


0.976 


2.9 


0.977 


0.977 


0.977 


0.978 


0.979 


0.979 


0.979 


0.980 


0.980 


0.981 


3.0 


0.981 


0.981 


0.982 


0.983 


0.983 


0.984 


0.984 


0.985 


0.985 


0.986 


3.1 


0.986 


0.987 


0.987 


0.987 


0.988 


0.988 


0.989 


0.990 


0.990 


0.990 


3.2 


0.991 


0.991 


0.992 


0.992 


0.992 


0.992 


0.994 


0.994 


0.994 


0.995 


3.3 


0.995 


0.996 


0.996 


0.997 


0.997 


0.998 


0.998 


0.999 


0.999 


0.999 


3.4 


1.000 


1.000 


1.000 


1.001 


1.002 


1.003 


1.003 


1.003 


1.004 


1.004 


3.5 


1.005 


1.005 


1.006 


1.006 


1.006 


1.007 


1.007 


1.008 


1.008 


1.009 


3.6 


1.009 


1.010 


1.010 


1.011 


1.011 


1.012 


1.012 


1.012 


1.013 


1.013 


3.7 


1.014 


1.014 


1.015 


1.015 


1.015 


1.016 


1.016 


1.017 


1.017 


1.017 


3.8 


1.018 


1.018 


1.019 


1.019 


1.019 


1.020 


1.021 


1.021 


1.022 


1.022 


3.9 


1.023 


1.023 


1.024 


1.024 


1.024 


1.025 


1.025 


1.025 


1.026 


1.026 


4.0 


1.027 


1.027 


1.028 


1.028 


1.029 


1.029 


1.030 


1.030 


1.030 


1.031 


■^v. 


0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


4 


1.027 


1.032 


1.036 


1.040 


1.044 


1.048 


1.053 


1.056 


1.060 


1.064 


5 


1.068 


1.073 


1.077 


1.080 


1.085 


1.089 


1.092 


1.097 


1.100 


1.103 


6 


1.107 


1.111 


1.115 


1.118 


1.121 


1.125 


1.128 


1.131 


1.135 


1.139 


7 


1.142 


1.146 


1.149 


1.152 


1.155 


1.159 


1.162 


1.164 


1.168 


1.171 


8 


1.174 


1.177 


1.180 


1.185 


1.187 


1.189 


1.193 


1.196 


1.199 


1.201 


9 


1.205 


1.207 


1.211 


1.213 


1.216 


1.218 


1.221 


1.225 


1.227 


1.230 


10 


1.233 


1.236 


1.239 


1.240 


1.243 


1.246 


1.249 


1.251 


1.253 


1.257 



*Unitsof sec /ft. 



3-63 



10 
9 
8 

7 
6 

5 



X 

< 



1.0 
0.9 
0.8 
0.7 

0.6 
0.5 

0.4 

0.3 



0.2 



^r"^] 1 1- 


. . n 


"~i~r^-^ 


I^'i7 ' W'irf^^'^Wi ''^ 


Ljm^^m'M 


^3E^= 




i 


WP^^iffi^ 


Wm^ 




^^ 


^:;: — - 


E— i ■A? 


li^J-^Jilp-^T^T^^ 


fUM^^^^ 


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0.1 
0.9999 0.999 0.9950.99 0.98 0.95 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 

K (Bretschneider, 1957) 



Figure 3-35. Relationship for Friction Loss Over a Bottom of Constant Depth 



3-64 



For the remaining part of this problem, either the value of F^„ equal 
to 220 nautical miles as determined from Figure 3-15 for U^ = 91.6, 
Hq = 59.4 can be used with the deepwater forecasting curves, or else 
Equation 3-40 can be used, as modified, 

H^ = 0.0555 U^ >JfJ + AF" 

along with Equation 3-36 

To = 2.13 V?^ . 



Fg is defined below. 

The latter being a numerical method is easier to use and more accurate 
than the graphical method of using forecasting curves . 

The procedure for computing wind waves over the Continental Shelf 
will be illustrated by using the bottom profile off the mouth of the 
Chesapeake Bay and the standard project hurricane developed for the 
Norfolk area. The storm surge computed for the standard project hurricane 
and 2.5 feet of astronomical tide are added to the mean low water depths 
to obtain the total water depth for wave generation. Refraction is 
neglected in this example, i.e., K^ = 1.0. The results of these computa- 
tions are given in Table 3-4 followed by examples and explanations. 

Column 1 of Table 3-4 is the distance in nautical miles measured 
seaward of the entrance to Chesapeake Bay, using increments of 5 nautical 
miles for each section. 

Column 2, d^, is the depth in feet referred to mean low water at the 
shoreward end of each section, denoted by X of Column 1. 

Colvimn 3 is the depth dj at the beginning of each section. 

Column 4 is the depth 6.2 at the shoreward end of each section. 
These depths are the water depths below MLW plus the 2.5-foot astro- 
nomical tide plus the hurricane surge and are then rounded off to the 
nearest foot. 

Column 5, d„, is the average of Columns 3 and 4 to the nearest foot. 

Column 6 is the effective fetch Fg (nautical miles), and is 
obtained for the first step directly from Equation 3-40. For successive 
steps, Fg = Fg + AF < 137 n.mi. where Fg is given in Column 14 one line 
above in each case (e.g., line X = 40, Fg = 80.6 + 5.0 = 85.6) and AF is 
5 n.mi. Fg is defined for Column 14. 



3-65 



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3-66 



Column 7 is the deepwater significant wave height H^ and is 
obtained from Equation 3-40; 

H^ = 0.0555 U^ y/F^ = 5.08 y/f^ , where U^ = 91.6 knots 

and F is obtained from Column 6. 

Column 8 is the deepwater significant wave period T^ and is 
obtained from Equation 3-36: 

T = 2.13 VH where H^ is obtained from Column 7. 

Column 9 is T^/dy, or Column 8 squared over Column 5. 

Column 10 is the shoaling coefficient (H/H' ) or K corresponding 
to the value of T^/dy, Column 9, and is obtained from Table 3-3 Kg 
versus T^/d. 

Column 11 is the friction loss parameter and is equal to 

frH K AX 0.01 H K, (5) (6,080) 304 H K 

M' M' M' 

where f^ is assumed as 0.01, AX = 5(6,080) = 30,400 ft., dj, is the 
average water depth of the increment AX. 

Column 12 is the friction factor K^ and is obtained from Figure 
3-35 where Ky- is a function of T^/dy (Column 9) and 

(fH K AX 

^ ,- , = A (Column 11). 

Column 13 is the equivalent deepwater wave height H^ and is 
obtained from H' = H K (the product of Columns 7 and 12) . 

Column 14 is the equivalent effective fetch length for H^ and is 
obtained from Equation 3-40 



F ' = 



H^ 



(0.0555 U^ ) 



H^ 



5.08 



where U^j = 91.6 knots (moving hurricane). 



Column 15 is obtained by using Equation 3-36 T^ = 2.13 vH^. 

Column 16 is (T')^/d„, where d„ is the water depth at shoreward 
end of section AX. 



3-67 



Column 17 is the shoaling coefficient Kg2 related to the values of 



(To) /da (Column 16). 

Column 18 is H = H^ x Kg2 (product of Columns 13 and 17) 
Column 19 is obtained by using Equation 3-38 



N = — = ' , where \ = R/Vp = — = 1.38 hours or 4,970 seconds. 
T/ Tj P 26 



Column 20 is H = 0.707 H /log N', H is from Column 18. 
max ^e 

After one line of computations across is completed, the next line is 
begun, using F = F' + AF <. 137 where F' is from Column 14 of the 
preceding completed line. For example, consider the line corresponding to 
X = 40 being completed. Then the computation for the next line X = 35 
is as follows: 

Fg = 66.4 from line X = 40, Column 14. 



Fg = 66.4 + 5 = 71.4 nautical miles for line X = 35 . 



Column 6 
Compute 

Column 7, H^ = 5.08 \ll\A'= 42.9 feet. 

Column 8, T^ = 2.13 \/42.9~ = 14.0 seconds . 

To (14)^ 

Column 9, =^ = = 1.73. 

dj, 113 



Column 10, K = 0.924 (Table 3-3) for values of J- = 1.73. 



304 HK (304 (42.9) (0.924) ^^^, 
Column 11, A = — _ = — rr7zzz~ = 0.943. 



(d.p)^ 



(113)^ 



Column 12, lO = 0.89 (from Figure 3-35) for values of =^ == 1.73 and 



Column 13, 
Column 14, 



A = 0.943. 
H^ = H^ Ky = 42.9 (0.89) = 38.2 feet . 



%- 



5.08 


2 


38.2' 
5.08 



= 56.6 nautical miles . 



3-68 



Column 15, t' = 2.13 >/38.2' = 13.2 seconds. 



Column 16, — = 1.58 
d- 



2 



Column 17, K^^ = 0.919. 

Column 18, H = 0.919 (38.2) = 35.1 feet, which is the shallow-water height 

for depth d^ = 110 feet, corresponding to MLW of 104 feet. 

4,970 , , , r 1- , , 

Column 19, N = ^ — = 377 or the total number or waves applicable to 

o 

steady-state significant wave of H = 35.1 feet, say 35 feet . 



Column 20, H_^ = 35.1(0.707) Vlog 377' = 60.4 feet, say 60 feet 



The moving fetch model of Wilson (1955) has been adapted for computer 
usage by Wilson (1961) . The basic equations were modified by Wilson (1966) . 
The Bretschneider (1959) model for hurricane wave prediction was modified 
by Bretschneider (1972) . Borgman (1972) used the results of Wilson (1957) 
to develop an approach for estimating the maximum wave in a storm which 
may be considered as an alternate to that presented here. 

3.8 WATER LEVEL FLUCTUATIONS 

The focus now changes from wave prediction to water level fluctuations 
in oceans and other bodies of water which have periods substantially longer 
than those associated with surface waves. Several known physical processes 
combine to cause these longer-term variations of the water level. 

The expression water level is used to indicate the mean elevation of 
the water when averaged over a period of time long enough (about 1 minute) 
to eliminate high frequency oscillations caused by surface gravity waves. 
In the discussion of gravity waves the water level was also referred to as 
the Stillwater level (SWL) to indicate the elevation of the water if all 
gravity waves were at rest. In the field, water levels are determined by 
measuring water surface elevations in a stilling well. Inflow and outflow 
of the well is restricted so that the rapid responses produced by gravity 
waves are filtered out, thus reflecting only the mean water elevation. 

Water level fluctuations -- classified by the characteristics and 
type of motion which take place — may be identified as: 

(a) astronomical tides 

(b) tsunamis 

(c) seiches 

(d) wave setup 

(e) storm surges 

(f) climatological variations 

(g) secular variations 



3-69 



The first five have periods that range from a few minutes to a few 
days; the last two have periods that range from semi-annually to many 
years. Although important in long-term changes in water elevations, 
climatological and secular variations are not discussed here. 

Forces caused by the graviational attraction between the moon, the 
sun, and the rotating earth result in periodic level changes in large 
bodies of water. The vertical rise and fall resulting from these forces 
is called the tide or astronomical tide; the horizontal movements of 
water are called tidal currents. The responses of water level changes to 
the tidal forces are modified in coastal regions because of variations in 
depths and lateral boundaries; tides vary substantially from place to 
place. Astronomical tide generating forces are well understood, and can 
be predicted many years in advance. The response to these forces can be 
determined from an analysis of tide gage records. Tide predictions are 
routinely made for many locations for which analyzed tide observations 
are available. In the United States, tide predictions are made by the 
National Ocean Survey, National Oceanographic and Atmospheric Administratioi 

Tsunamis are generated by several mechanisms: submarine earthquakes, 
submarine landslides, and underwater volcanos. These waves may travel 
distances of more than 5,000 miles across an ocean with speeds at times 
exceeding 500 miles per hour. In open oceans, the heights of these waves 
are generally unknown but small; heights in coastal regions have been 
greater than 100 feet . 

Seiches are long-period standing waves that continue after the 
forces that start them have ceased to act. They occur commonly in 
enclosed or partially enclosed basins. 

Wave setup is defined as the superelevation of the water surface due 
to the onshore mass transport of the water by wave action alone. Isolated 
observations have shown that wave setup does occur in the surf zone. 

Surges are caused by moving atmospheric pressure jumps and by the 
wind stress accompanying moving storm systems. Storm systems are signifi- 
cant because of their frequency and potential for causing abnormal water 
levels at coastlines. In many coastal regions, maximum storm surges are 
produced by severe tropical cyclones called hurricanes. 

Prediction of water level changes is complex because many types of 
water level fluctuations can occur simultaneously. It is not unusual 
for surface wave setup, high astronomical tides, and storm surge to occur 
coincidently at the shore on the open coast. It is difficult to determine 
how much rise can be attributed to each of these causes. Although astro- 
nomical tides can be predicted rather well where levels have been recorded 
for a year or more, there are many locations where this information is not 
available. Furthermore, the interaction between tides and storm surge in 
shallow water is not well defined. 



3-70 



3.81 ASTRONOMICAL TIDES 

Tide is a periodic rising and falling of sea level caused by the 
gravitational attraction of the moon, sun, and other astronomical bodies 
acting on the rotating earth. Tides follow the moon more closely than they 
do the sun. There are usually two high and and two low waters in a tidal 
or lunar day. As the lunar day is about 50 minutes longer than the solar 
day, tides occur about 50 minutes later each day. Typical tide curves for 
various locations along the Atlantic, Gulf, and Pacific coasts of the 
United States are shown in Figures 3-36 and 3-37. Along the Atlantic 
coast, the two tides each day are of nearly the same height. On the Gulf 
coast, the tides are low but in some instances have a pronounced diurnal 
inequality. Pacific coast tides compare in height with those on the 
Atlantic coast but in most cases have a decided diurnal inequality. 
(See Appendix A, Figure A-10.) 

The dynamic theory of tides was formulated by Laplace (1775) and 
special solutions have been obtained by Doodson and Warburg (1941) among 
others. The use of simplified theories for the analysis and prediction 
of tides has been described by Schureman (1941), Defant (1961) and Ippen 
(1966) . The computer program for tide prediction, currently being used 
for official tide prediction in the United States is described by Pore 
and Cummings (1967). 

Data concerning tidal ranges along the seacoasts of the United States 
are given to the nearest foot in Table 3-5. Spring ranges are shown for 
areas having approximately equal daily tides; diurnal ranges are shown 
for areas having either a diurnal tide or a pronounced diurnal inequality. 
Detailed data concerning tidal ranges are published annually in Tide Tables, 
U.S. Department of Commerce, National Ocean Survey. 

3.82 TSUNAMIS 

Long period gravity waves generated by such disturbances as earth- 
quakes, landslides, volcano eruptions and explosions near the sea surface 
are called tsunamis. The Japanese word tsunami has been adopted to replace 
the expression tidal wave to avoid confusion with the astronomical tides. 

Most tsunamis are caused by earthquakes that extend at least partly 
under the sea, although not all submarine earthquakes produce tsunamis. 
Severe tsunamis are rare events. 

Tsunamis may be compared to the wave generated by dropping a rock in 
a pond. Waves (ripples) move outward from the source region in every 
direction. In general, the tsunami wave amplitudes decrease but the 
number of individual waves increases with distance from the source region. 
Tsunami waves may be reflected, refracted, or diffracted by islands, sea- 
mounts, submarine ridges or shores. The longest waves travel across the 
deepest part of the sea as shallow-water waves, and may obtain speeds of 
several hundred knots. The travel time required for the first tsunami 
disturbance to arrive at any location can be determined within a few 
percent of the actual travel time by the use of suitable tsunami travel- 
time charts. 



3-71 



DAr 10 II 12 13 



14 15 16 17 



18 19 




LuMf dita: mu. S. declirntion, 9th; ipogee, lOlb: lul quirter, 13th: on equitor, 16th; new moon. 20Ui; peri|M, 
22d: mu. N. dcdinition, 23d. 

(from Notionol Ocean Survey, NOAA, Tide Tables) 

Figure 3-36. Typical Tide Curves Along Atlantic and 
Gulf Coasts 



3-72 



DAY 



Ft 

6 
4 
2 

-2 



12 

10 

8 

6 

4 

2 



-2 



18 

16 

14 

12 

10 

8 

6 

4 

2 



-2 



32 
30 
28 
26 
24 
22 
20 
18 
16 
14 
12 
10 



-2 



10 11 



^^^^^^^^^^^ 



12 13 14 15 16 



SAN FRANCISCO 



17 18 19 20 



SEATTLE 




lunat (Jala: max S. declination, 9th, apogee, lOlh, last quartet, 13tli; on equalor, 16lh, new moon. 20th, perigee, 
22d, max. N. declination, 23d. 

(from Notional Ocean Survey, NOAA, Tide Tobies) 

Figure 3-37. Typical Tide Curves Along Pacific Coasts of the 
United States 



3-73 



Table 3-5. Tidal Ranges 



Station 


Approximate Ranges (feet) 


Mean 


Diurnal 


Spring 


Atlantic Coast 








Calais, Maine 


20 




23 


W. Quoddy Head, Maine 


16 




18 


Englishman Bay, Maine 


12 




14 


Belfast, Maine 


10 




11 


Provincetown, Mass. 


9 




11 


Chatham, Mass. 


7 




8 


Cut t yhunk , Mas s . 


3 




4 


Saybrook, Conn. 


4 




4 


Montauk Point, N.Y. 


2 




2 


Sandy Hook, N.J. 


5 




6 


Cape May, N.J. 


4 




5 


Cape Henry, Va. 


3 




3 


Charleston, S.C. 


5 




6 


Savannah, Ga. 


7 




8 


Mayport, Fla. 


5 




5 


Gulf Coast 








Key West, Fla 


1 




2 


Apalachicola, Fla. 




1 




Atchafalaya Bay, La. 


1 


2 




Port Isabel, Tex. 


1 


1 




Pacific Coast 


Point Loma, Calif. 


4 


5 




Cape Mendocino, Calif. 


4 


6 




Siuslaw River, Ore. 


5 


7 




Columbia River, Wash. 


6 


8 




Port Townsend, Wash. 


7 


10 




Puget Sound, Wash. 


11 


15 





Tsunamis cross the sea as very long waves of low amplitude. A wave- 
length of 100 miles and an amplitude of 2 feet is not unreasonable. The 
wave may be greatly amplified by shoaling, diffraction, convergence, and 
resonance when it reaches land. Sea water has been carried higher than 
35 feet above sea level in Hilo, Hawaii by tsunamis. Tide gage records 
of the tsunami of 23-26 May 1960 at these locations are shown in Figure 
3-38. The tsunami appears as a quasi-periodic oscillation, superimposed 
on the normal tide. The characteristic period of the disturbance, as well 
as the amplitude, is different at each of the three locations. It is 
generally assumed that the recorded disturbance results from forced oscil- 
lations of hydraulic basin systems, and that the periods of greatest re- 
sponse are determined by basin geometry. 



3-74 




Tide Gage Record Showing Tsunami 

HONOLULU. HAWAII 

May 23-24, 1960 




Approx. Hours G.M.T. 

12 13 14 15 16 17 18 19 20 21 

_l \ I I I I I I I l_ 



Tide Gage Record Showing Tsunami . 
MOKUOLOE ISLAND. HAWAII 
May 23-24. 1960 




Tide Gage Record Showing Tsunami 

JOHNSTON ISLAND, HAWAII 

May 23-24. 1960 



A-^X. 




(from Symons and Zelter, I960) 
Figure 3-38. Sample Tsunami Records from Tide Gages 



3-75 



Theoretical and applied research dealing with tsunami problems has 
been greatly intensified since 1960. Preisendorfer (1971) lists more than 
60 significant theoretical papers published since 1960. The list does not 
include observational papers concerned with the warning system. 

3.83 LAKE LEVELS 

Lakes have insignificant tidal variations, but are subject to sea- 
sonal and annual hydrologic changes in water level and to water level 
changes caused by wind setup, barometric pressure variations, and seiches. 
Additionally some lakes are subject to occasional water level changes by 
regulatory control works. 

Water surface elevations of the Great Lakes vary irregularly from 
year to year. During each year, the water surfaces consistently fall to 
their lowest stages during the winter and rise to their highest stages 
during the summer. Nearly all precipitation in the watershed areas during 
the winter is snow or rainfall transformed to ice. When the temperature 
begins to rise there is substantial runoff - thus the higher stages in the 
summer. Typical seasonal and yearly changes in water levels for Lake Erie 
are shown in Figure 3-39. The maximum and minimum monthly mean stages for 
the lakes are summarized in Table 3-6. 

Table 3-6. Fluctuations in Water Levels - Great Lakes System (1860 through 1973). 



Lake 


Datum* 
Factor 


Lowt 
Water 
Chart 
Datum 


Surface 
Elevation 


Alltime Monthly Means 


High 
(feet) 


Date 


Low 

(feet) 


Date 


Difference 


Superior 

Michigan-Huron 

St. Clair t 

Erie 

Ontario 


L71 
L74 
L82 
L94 

1.23 


600.0 
576.8 
571.7 
568.6 
242.8 


600.37 
578.68 
573.01 
570.38 

244.77 


602.06 
581.94 
576.22 
573.52 
248.06 


8/1876 
6/1886 
6/1973 
6/1973 
6/1952 


598.23 
575.35 
569.86 
567.49 
241.45 


4/1926 
3/1964 
1/1936 
2/1936 
11/1934 


3.83 
6.59 
6.36 
6.03 
6.61 



Elevations are in feet above mean water level at Father Point, Quebec. 

International Great Lakes Datum (IGLD) (1955). 

* To convert to U.S. Lake Survey 1935 Datum, add datum factor to IGLD (USLS 1935 = IGLD + datum factor). 

t Low water datum is the zero plane on Lake Survey Charts to which charts are referred. Thus the zero (low water) 

datum on a USLS Lake Superior chart is 600 feet above mean waterlevel at Father Point, Quebec. 
If Lake St. Clair elevations are available only for 1898 to date. 

In addition to seasonal and annual fluctuations, the Great Lakes are 
subject to occasional seiches of irregular amount and duration. These 
sometimes result from a resonant coupling which arises when the propaga- 
tion speed of an atmospheric disturbance is nearly equal to the speed of 
free waves on a lake. (Ewing, Press and Donn, 1954), (Harris, 1957), 
Platzman, 1958, 1965.) The lakes, also and sometimes simultaneously, are 
affected by wind stresses which raise the water level at one end and lower 
it at the other. These mechanisms may produce changes in water elevation 
ranging from a few inches to more than 6 feet. Lake Erie, shallowest of 



3-76 



1 , 1 Nnr 


o 

00 
O 


S \ . T " ^"' 




~ -3 X.A-_^-_— _ 


^i \^. _ - -1. - 


■» 1 r " •" 


__ ;T--^ r~ 7 " "" » 

E 1 1 1 S 330 


■S / ' ' S AON 


i-.-,^-— -//- n---i "° 


Lj/.___,^,:: ,__J - 


?_/ ___y.^ 1 ="" 


<T> 2? V ' CO - 


-" ~ , "«>■ 


;$r:--|- ^-^----^--------^|--j ^,. 


"^jS 5 \ \- j 


lifS s N ^^, s 




^ o i "^ 1 > 3 1 Nvr 


4_L^ i 1 1 E f ? =30 


__M = > /. 5 ? 


? /z^' I i 


^ /// ^ ? 


i ^77 "1 on. 




S 1 S NflP 


s 1 \ .^ 1 " ■^"'^ 






i t ' , ^ 5 83i 


S L 1 7 Nvr 



o) Z. — a> 



a> 


> 


o» 


> 


•o 


~ 






a> 


a> 




O) 


o 


o 




6 


(^ 


w 


^" 


^ 




» 


K 


« 


q: 


< 


LU 


UJ 






C 
o 



> 



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a> 

u 
(1> 
+J 



o 

•H 



I 

■H 



in 
uunjDQ 



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»3- ro OJ — O 



3-77 



the Great Lakes, is subject to greater wind-induced surface fluctuations, 
that is, wind setup, than any other Lake. Wind setup is discussed in 
Section 3.86, Storm Surge and Wind Setup. 

In general, the maximum amount of these irregular changes in lake 
level must be determined for each location under consideration. Table 3-7 
shows short-period observed maximum and minimum water level elevations at 
selected gage sites. More detailed data on seasonal lake levels and wind 
setup may be obtained for specific locations from the Lake Survey Center, 
National Oceanic and Atmospheric Administration, U.S. Department of 
Commerce. 

Table 3-7. Short-Period Fluctuations in Lake Levels at Selected Gage Sites 



Lake and Gage Location 


Period of 


Maximum 


Recorded 




Gage Record 


Rise in Feet 


Fall in Feet 






(MSE)* 


(MSE)* 


SUPERIOR at Marquette 


1903-1971 


3.7 


2.9 


MICHIGAN at Calument Harbor 


1903-1971 


4.5 


5.2 


(Chicago) 








HURON at Harbor Beach 


1902-1971 


3.3 


4.5 


ERIE at Buffalo 


1900-1971 


8.7 


6.2 


ERIE at Toledo 


1940-1971 


5.3 


8.9 


ONTARIO at Oswego 


1933-1971 


4.2 


3.8 



*The mean surface elevation (MSE) refers to the water level that represents 
the average water elevation for the period of record. It corresponds to 
the mean surface elevation given in Table 3-6. 

3.84 SEICHES 

Seiches are standing waves (Fig. 3-40) of relatively long periods 
that occur in lakes, canals, bays and along open seacoasts. Lake seiches 
are usually the result of a sudden change, or a series of intermittent- 
periodic changes, in atmospheric pressure or wind velocity. Standing 
waves in canals can be initiated by suddenly adding or subtracting large 
quantities of water. Seiches in bays can be generated by local changes 
in atmospheric pressure and wind and by oscillations transmitted through 
the mouth of the bay from the open sea. Open-sea seiches can be caused 
by changes in atmospheric pressure and wind, or tsunamis. Standing waves 
of large amplitude are likely to be generated if the causative force which 
sets the water basin in motion is periodic in character, especially if the 
period of this force is the same as, or is in resonance with, the natural 
or free oscillating period of the basin. (See Section 2.5, Wave Reflection.) 



3-78 



Free oscillations have periods that are dependent upon the horizontal 
and vertical dimensions of the basin, the number of nodes of the standing 
wave, that is, lines where deviation of the free surface from its undis- 
turbed value is zero, and friction, ihe period of a true forced-wave 
oscillation is the same as the period of the causative force. Forced 
oscillations, however, are usually generated by intermittent external 
forces, and the period of the oscillation is determined partly by the 
period of the external force and partly by the dimensions of the water 
basin and the mode of oscillation. Oscillations of this type have been 
called forced seiches (Chrystal, 1905) to distinguish them from free 
seiches in which the oscillations are free. 











f 






^V^ 








^^^ 










'^ 


YV 


k 


A 


/ 


aJ\.>\J\ 


(a) Profile 


+ -w+J+« + 


t+ 


•-»4 


u- 


-i 



(b) Water Motion 



(A) STANDING WAVES 



(B) CLOSED BASIN 

(I). Fundomentol Mode 
(First Harmonic) 




Node jf 
-Antinodes 

(2). Second Mode 

(Second Hormonic) 




(3)Third Mode 

(Third Harmonic) 










(C) OPEN-ENDED BASIN 

(I). Fundamental Mode 
(First Harmonic) 

L„ 



h 



(2). Second Mode 

(Third Harmonic) 




(3).Third Mode 

(Fifth Harmonic) 




Surface profiles for oscllloting waves 

(after CQrr,l953) 
Figure 3-40. Long- Wave Surface Profiles 

For the simplest form of a standing one-dimensional wave in a closed 
rectangular basin with vertical sides and uniform depth (Fig. 3-40(B)), 
wave antinodes, that is, lines where deviation of the free surface from 
its undisturbed value is a relative maxima or minima, are situated at the 
ends (longitudinal seiche) or sides (transverse seiche). The number of 
nodes and antinodes in a basin depends on which mode or modes of oscilla- 
tion are present. If n = number of nodes along a given basin axis, d = 
basin depth, and £g = basin length along that axis, then Tn, the natural 
free oscillating period is given by 



2f 



T„ = 



B 



nVgdT 



(3-42) 



3-79 



The fundamental and maximum period (T^ for n = 1) becomes 

If 
T =— ^ (3-43) 

Equation 3-43 is called Merian's formula. (Sverdrup, Johnson and Fleming, 
1942.) 

In an open rectangular basin of length %' and constant depth d, 
the simplest form of a one-dimensional, non-resonant, standing longitudi- 
nal wave is one with a node at the opening, antinode at the opposite end, 
and n' nodes in between. (See Figure 3-40(C).) The free oscillation 
period T^/ in this case is 

4V 



For the fundamental mode (n' = 0), T^ becomes 

T' =^1mL (3-45) 

The basin's total length is occupied by one-fourth of a wave length. 

This simplified theory must be modified for most actual basins, 
because of the variation in width and depth along the basin axes. 

Defant (1961) outlines a method to determine the possible periods for 
one-dimensional free oscillations in long narrow lakes of variable width 
and depth. Defant 's method is useful in engineering work because it permits 
computation of periods of oscillation, relative magnitudes of the vertical 
displacements along chosen axes, and the positions of nodal and antinodal 
lines. This method, applicable only to free oscillations, can be used to 
determine the nodes of oscillation of multinodal and uninodal seiches. 
The theory for a particular forced oscillation was also derived by Defant 
and is discussed by Sverdrup et al (1942). Hunt (1959) discusses some 
complexities involved in the hydraulic problems of Lake Erie, and offers 
an interim solution to the problem of vertical displacement of water at 
the eastern end of the lake. More recently, work has been done by Simpson 
and Anderson (1964), Platzman and Rao (1963), and Mortimer (1965). 
Rockwell (1966) computed the first five modes of oscillation for each of 
the Great Lakes by a procedure based on the work of Platzman and Rao (1965). 
Platzman (1972) has developed a method for evaluating natural periods for 
basins of general two-dimensional configuration. 

3.85 WAVE SETUP 

Field observations indicate that part of the variation in mean water 
level near shore is a function of the incoming wave field. However these 



3-80 



observations are insufficient to provide quantitative trends. (Savage, 
1957; Fairchild, 1958; Dorrestein, 1962; Galvin and Eagleson, 1965.) A 
laboratory study by Saville (1961) indicated that for waves breaking on 
a slope there would be a decrease in the mean water level relative to the 
Stillwater level just prior to breaking, with a maximum depression or set- 
down at about the breaking point. This study also indicated that from the 
breaking point the mean water surface slopes upward to the point of inter- 
section with the shore and has been termed wave setup. Wcwe setup is 
defined as that super -elevation of the mean water level caused by wave 
action alone. This phenomenon is related to a conversion of kinetic 
energy of wave motion to a quasi-steady potential energy. 

Theoretical studies of wave setup have been made by Dorrestein (1962), 
Fortak (1962), Longuet-Higgins and Stewart (1960, 1962, 1963, 1964), Bowen, 
Inman, and Simmons (1968), and Hwang and Divoky (1970). Theoretical devel- 
opments can account for many of the principal processes, but contain 
factors that are often difficult to specify in practical problems. 

R.O. Reid (personal communication) has suggested the following approach 
for estimating the wave setup at shore, using Longuet-Higgins and Stewart 
(1963) theory for the setdown at the breaking zone and solitary wave theory. 
The theory for setdown at the breaking zone indicates that 

gi/2 Hj T 

in which Sj^ is the setdown at the breaking zone, T is the wave period, 
Hq is the deepwater significant wave height, dj^ is the depth of water 

at the breaker point and g is gravity. The laboratory data of Saville 
(1961) gives somewhat larger values than those obtained by use of Equation 
3-46. 

By using relations derived from solitary wave theory relating dh 
to the breaker height of the significant wave, H^j, and d^/H^ to Ho/Lo, 
the above relation can be converted to 

3/2 



0.536 H, 
Su = — -^— . (3-47) 



'b 



g 



1/2 J 



Longuet-Higgins and Stewart (1963) show from an analysis of Saville's 
data that the wave setup AS between the breaker zone and shore is given 
approximately by AS = 0.15 dj. Assuming that djy = 1.28 Hj,, this becomes 
AS = 0.19 Hfe. 

The net wave setup at the shore is 

Sj^, = AS + Sj, , (3-48) 



or 



Sh; = 0.19 



1 - 2.82 



3-81 



gT^ 



Hy . (3-49) 



Equation 3-49 provides a conservative estimate for wave setup at the 
shore. The difference between laboratory data and theory, however, is not 
likely to exceed the uncertainties of field data. 



************** 



EXAMPLE PROBLEM ************** 



GIVEN : Hh = 20 feet, T = 12 seconds. 
FIND : Wave setup, Sw 
SOLUTION : Using Equation 3-49, 



S^ = 0.19 



Sj^ = 0.19 



gTV J 


Hfc, 


20 Y^ 


32.2 (12)y 



1- 2.82 



1-2.82 20 



Sj^; = 3.1 feet . Say 3 feet . 
Equation 3-49 is only applicable to normal beach slopes. 
************************************* 

3.86 STORM SURGE AND WIND SETUP 

3.861 General . Reliable estimates of water-level changes under storm 
conditions are essential for the planning and design of coastal engineer- 
ing works. Determination of design water elevations during storms is a 
complex problem involving interaction between wind and water, differences 
in atmospheric pressure, and effects caused by other mechanisms unrelated 
to the storm. Winds are responsible for the largest changes in water 
level when considering only the storm-surge generating processes. A wind 
blowing over a body of water exerts a horizontal force on the water surface 
and induces a surface current in the general direction of the wind. The 
force of wind on the water is partly due to inequalities of air pressures 
on the windward side of gravity waves, and partly due to shearing stresses 
at the water surface. Horizontal currents induced by the wind are impeded 
in shallow water areas, thus causing the water level to rise downwind 
while at the windward side the water level falls. The term storm surge is 
used to indicate departure from normal water level due to the action of 
storms. The term wind setup is often used to indicate rises in lakes, 
reservoirs and smaller bodies of water. A fall of water level below the 
normal level at the upwind side of a basin is generally referred to as 
setdown. 

Severe storms may produce surges in excess of 25 feet on the open 
coast and even higher in bays and estuaries. Generally, setups in lakes 
and reservoirs are less, and setdown in these enclosed basins is about 
equivalent to the setup. Setdown in open oceans is insignificant because 
the volume of water required to produce the setup along the shallow regions 
of the coast is small compared to the volume of water in the ocean. How- 
ever, setdown may be appreciable when a storm traverses a relatively narrow 



3-82 



landmass such as southern Florida and moves offshore. High offshore winds 
in this case can cause the water level to drop several feet. 

Setdown in semienclosed basins (bays and estuaries) also may be sub- 
stantial, but the fall in water level is influenced by the coupling to the 
sea. There are some detrimental effects as a result of setdown, such as 
making water-pumping facilities inoperable due to exposure of the intake, 
increasing the pumping heads of such facilities, and causing navigational 
hazards because of decreased depths. 

However, rises in water levels (setup rather than setdown) are of 
most concern. Abnormal rises in water level in nearshore regions will not 
only flood low-lying terrain, but provide a base on which high waves can 
attack the upper part of a beach and penetrate farther inland. Flooding 
of this type combined with the action of surface waves can cause severe 
damage to low- lying land and backshore improvements. 

Wind-induced surge, accompanied by wave action, accounts for most of 
the damage to coastal engineering works and beach areas. Displacement of 
stone armor units of jetties, groins and breakwaters, scouring around 
structures, accretion and erosion of beach materials, cutting of new in- 
lets through barrier beaches, and shoaling of navigational channels can 
often be attributed to storm surge and surface waves. Moreover, surge can 
increase hazards to navigation, impede vessel traffic, and hamper harbor 
operations. A knowledge of the increase and decrease in water levels 
that can be expected during the life of a coastal structure or project is 
necessary to design structures that will remain functional. 

3.862 Storms . A storm is an atmospheric disturbance characterized by 
high winds which may or may not be accompanied by precipitation. Two 
distinctions are made in classifying storms: a storm originating in the 
tropics is called a tropical storm; a storm resulting from a cold and 
warm front is called an extratropioat storm. Both of these storms can 
produce abnormal rises in water level in shallow water near the edge of 
water bodies. The highest water levels produced along the entire gulf 
coast and from Cape Cod to the south tip of Florida on the east coast 
generally result from tropical storms. High water levels are rarely 
caused by tropical storms on the lower coast of California. Extreme 
water levels in some enclosed bodies, such as Lake Okeechobee, Florida 
can also be caused by a tropical storm. Highest water levels at other 
coastal locations and most enclosed bodies of water result from extra- 
tropical storms. 

A severe tropical storm is called a hurricane when the maximum 
sustained wind speeds reach 75 miles per hour (65 knots). Hurricane 
winds may reach sustained speeds of more than 150 miles per hour (130 
knots). Hiirricanes, unlike less severe tropical storms, generally are 
well organized and have a circular wind pattern with winds revolving 
around a center or eye (not necessarily the geometric center). The eye 
is an area of low atmospheric pressure and light winds. Atmospheric 
pressure and wind speed increase rapidly with distance outward from the 
eye to a zone of maximum wind speed which may be anywhere from 4 to 60 



3-83 



nautical miles from the center. From the zone of maximum wind to the 
periphery of the hurricane, the pressure continues to increase; however, 
the wind speed decreases. The atmospheric pressure within the eye is the 
best single index for estimating the surge potential of a hurricane. This 
pressure is referred to as the centvdl pressure index (CPI) . Generally 
for hurricanes of fixed size, the lower the CPI, the higher the wind speeds, 
Hurricanes may also be characterized by other important elements, such as 
the radius of maximum winds (R) which is an index of the size of the storm, 
and the speed of forward motion of the storm system (Vj.) . A discussion of 
the formation, development and general characteristics of hurricanes is 
given by Dunn and Miller (1964) . 

Extratropical storms that occur along the northern part of the east 
coast of the United States accompanied by strong winds blowing from the 
northeast quadrant are called northeasters. Nearly all destructive north- 
easters have occurred in the period from November to April; the hurricane 
season is from about June to November. A typical northeaster consists of 
a single center of low pressure and the winds revolve about this center, 
but wind patterns are less symmetrical than those associated with hurri- 
canes . 

3.863 Factors of Storm Surge Generation . The extent to which water 
levels will depart from normal during a storm depends on several factors. 
The factors are related to the: 

(a) characteristics and behavior of the storm; 

(b) hydrography of the basin; 

(c) initial state of the system; and 

(d) other effects that can be considered external to the system. 

Several distinct factors that may be responsible for changing water levels 
during the passage of a storm may be identified as: 

(a) astronomical tides 

(b) direct winds 

(c) atmospheric pressure differences 

(d) earth's rotation 

(e) rainfall 

(f) surface waves and associated wave setup 

(g) storm motion effects . 

The elevation of setup or setdown in a basin depends on storm inten- 
sity, path or track, overwater duration, atmospheric pressure variation, 
speed of translation, storm size, and associated rainfall. Basin charac- 
teristics that influence water-level changes are basin size and shape, 
and bottom configuration and roughness. The size of the storm relative 
to the size of the basin is also important. The magnitude of storm 
surges is shown in Figures 3-41 and 3-42. Figure 3-41 shows the differ- 
ence between observed water levels and predicted astronomical tide levels 
during Hurricane Carla (1961) at several Texas and Louisiana coast tide 
stations. Figure 3-42 shows high water marks obtained from a storm survey 
made after Hurricane Carla. Harris (1963b) gives similar data from other 
hurricanes . 



3-84 



3.864 Initial Water Level . Water surfaces on the open coast or in en- 
closed or semienclosed basins are not always at their normal level prior 
to the arrival of a storm. This departure of the water surface from its 
normal position in the absence of astronomical tides, referred to as an 
initial water level, is a contributing factor to the water level reached 
during the passage of a storm system. This level may be 2 feet above 
normal for some locations along the U.S. Gulf coast. Some writers refer 
to this difference in water level as a forerunner in advance of the storm 
due to initial circulation and water transport by waves particularly when 
the water level is above normal. Harris (1963b) on the other hand, indi- 
cates that this general rise may be due to short-period anomalies in the 
mean sea level not related to hurricanes. Whatever the cause, the initial 
water level should be considered when evaluating the components of open- 
coast storm surge. The existence of an initial water level preceeding the 
approach of Hurricane Carla is shown in Figure 3-41 and in a study of the 
synoptic weather charts for this storm. (Harris, 1963b.) At 0700 hours 
(Eastern Standard Time) 9 September 1961, the winds at Galveston, Texas 
were about 10 mph, but the open coast tide station (Pleasure Pier) shows 
the difference between the observed water level and astronomical tide to 
be above 2 feet. Rises of this nature on the open coast can also affect 
levels in bays and estuaries. 

There are other causes for departures of the water levels from normal 
in semienclosed and enclosed basins, such as the effects of evaporation 
and rainfall. Generally, rainfall plays a more dominant role since these 
basins are affected by direct rainfall and can be greatly affected by 
rainfall runoff from rivers. The initial rise caused by rainfall is due 
to rains preceding the storm; rains during the passage of a storm have a 
time-dependent effect on the change in water level. 

3.865 Storm Surge Prediction . The design of coastal engineering works is 
usually based on a life expectancy for the project and on the degree of 
protection the project is expected to provide. This design requires that 
the design storm have a specified frequency of occurrence for the partic- 
ular area. An estimate of the frequency of occurrence of a particular 
storm surge is required. One method of making this estimate is to use 
frequency curves developed from statistical analyses of historical water 
level data. Table 3-8, based on National Ocean Survey tide gage records, 
indicates observed extreme storm surge water levels including wave setup 
The water levels are those actually recorded at the various tide stations, 
and do not necessarily reflect the extreme water levels that may have 
occurred near the gages. Values in this table may differ from gage-station 
values because of corrections for seasonal and secular anaomalies. The 
frequency of occurrence for the highest and lowest water levels may be 
estimated by noting the length of time over which observations were made. 
The average yearly highest water level is an average of the highest water 
level from each year during the period of observation. Extreme water 
levels are rarely recorded by water level gages, partly because the gages 
tend to become inoperative with extremely high waves, and partly because 
the peak storm surge often occurs between tide gage stations. Post-storm 
surveys showed water levels, as the result of Hurricane Camille, August 



3-85 



90* 




Fort Point 

Pier 21 
Pleoture Pier 



-40° 




«S»J 



40" — 



(from Harris, 1963 b) 

Figure 3-41. Storm Surge and Observed Tide Chart. Hurricane C:irla, 
7-12 September 1961. Insert Maps for Freeport and 
Galveston, Texas, Areas 



3-86 




Time Eye of Hurricane 
Neorest to Stotion 



September, 1961 



j 


r; 






' 








^ 


r I I 

Oft Foiled 




1 


















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96 
1 


























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, Po 


n 


Con 


or ^ 




^ 


r 
















-. 


^ 






'^ 


/ 
























c< 


lOfO 


lo River 


.oc 


t(K 


ata 


6or< 


cl 

















September, 1961 





r~ 


n 


r 










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^1 


1,7 


— 1 






"^ 


















J 




\l 


A. 


(•sir 


g D< 


to 


















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10.9 


> 




















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/a 


.9 




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Brotorio Co. j 




, . 


d^ 


Gogfl 

Follfld — 


^ 


r^ 




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Dow Chemicor'a! 


^y 


rJ 


y^ 


^ 




V 




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-^ 


Boies' 


L^-a;;] 


,/ 


^ 


y 










V 


-^ 








r- 


-^ 


X^ 


/ 


















\ 






Brazos'Rlve'r Gates 

i 1 1 1 






















V 



+ 2(— 
1 - 

ScoU 




m-m 



September, 1961 



September, 1961 




Septetuber, 1961 



Figure 3-41. Storm Surge and Observed Tide Chart. Hurricane Carla, 
7-12 September 1961. Insert Maps for Freeport and 
Galveston, Texas, Areas -- Continued 



3-87 







CM 






^ 


..^^ 




^ — 




1 


^M/^^ 






«| 







u 

E 
■»-> 

Oh 



u 




<a 




u 


• 




M 


0) 


C 


c 


■H 


OS 


TJ 


o 


O 


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o 


fH 


1— 1 


fH 


^-^ 


3 




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tw 




o 


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to 


c 


rt 


(U 


X 


+-> 


a> 


X 


H 


a> 


^ 


0) 


o 


^ 


it-i 


+j 


■»-> 


in 


iH 


<i> 


rt 


4-> 


X 


a 


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o 




•H 


^ 


T5 


^^ 


c 


nt 


•H 


:?: 






cS 


u 


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o 


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rt 


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a: 


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to 




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PL, 



3-88 





Table 3-8. Highest and Lowest Water Levels 








Observation 
Period 


Mean 
Range 
(feet) 


Highest Water Leveb 

Above Mean High Water 

(feet) 


Lowest Water Levels 

Below Low Water 

(feet) 


Location 


Average 
Yearly 
Highest 


Extreme 
High 


Date of 
Record 


Average 
Yearly 
Lowest 


Extreme 
Low 


Date of 
Record 


ATLANTIC COAST 


















Eastport, Maine 


1930-69 


18.2 


3.9 


4.9 


21 Dec 68 


3.9 


4.4 


7 Jan 43 
23 May 59 
30 Dec 63 


Portland, Maine 


1912-69 


9.0 


2.8 


4.0 


30 Nov 44 
20 Nov 45 


2.8 


3.7 


30 Nov 55 


Bar Harbor, Maine 


1947-70 


10.5 


3.1 


4.3 


29 Dec 59 


2.8 


3.6 


30 Dec 53 


Portsmouth, N.H. 


1927-70 


8.1 


2.8 


3.7 


30 Nov 44 
29 Dec 59 


2.7 


3.4 


30 Nov 55 


Boston, Mass. 


1922-70 


9.5 


3.0 


4.4 


29 Dec 59 


3.1 


3.8 


25 Jan 28 

24 Mar 40 


Woods Hole, Mass. 


1933-70 


1.8 


2.9 


9.2 


21 Sep 38 


2.0 


2.7 


8 Jan 68 


Providence, R.L 


1938-47 
1957-70 


4.6 


3.7 


13.1 


21 Sep 38 


2.5 


3.4 


5 Jan 59 


Newport, R.I. 


1931-70 


3.5 


2.7 


10.0 


21 Sep 38 


2.1 


2.9 


25 Jan 36 


New London, Conn. 


1938-70 


2.6 


3.2 


8.1 


21 Sep 38 


2.3 


3.4 


11 Dec 43 


Willets Point, N.Y. 


1932-70 


7.1 


3.8 


9.6 


21 Sep 38 


3.2 


4.1 


24 Mar 40 


Battery, N.Y. 


1920-70 


4.5 


2.9 


5.7 


12 Sep 60 


3.0 


4.0 


8 Mar 32 
5 Jan 59 


Montauk, N.Y. 


1948-70 


2.1 


3.2 


6.6 


31 Aug 54 


1.9 


2.6 


8 Feb 51 


Sandy Hook, N.J. 


1933-70 


4.6 


3.1 


5.7 


12 Sep 60 


2.9 


4.1 


31 Dec 62 


Atlantic City, N.J. 


1912-20 
1923-70 


4.1 


2.8 


5.2 


14 Sep 44 


2.7 


3.7 


8 Mar 32 


Philadelphia, Pa. 


1900-20 
1922-70 


5.9 


2.2 


4.7 


25 Nov 50 


3.1 


6.7 


31 Dec 62 


Lewes, Del. 
Breakwater Harbor 


1936-70 


4.1 


3.0 


5.4 


6 Mar 62 


2.5 


3.0 


28 Mar 55 


Baltimore, Md. 


1902-70 


1.1 


2.3 


6.7 


23 Aug 33 


3.4 


5.0 


24 Jan 08 


Annapolis, Md. 


1929-70 


0.9 


2.2 


5.4 


23 Aug 33 


2.8 


3.8 


31 Dec 62 


Solomons Island, Md. 


1938-70 


1.2 


2.0 


3.4 


13 Aug 55 


2.1 


3.4 


31 Dec 62 


Washington, D.C. 


1931-70 


2.9 


2.8 


8.4 


17 Oct 42 


2.9 


3.9 


31 Dec 62 


Portsmouth, Va. 


1935-70 


2.8 


3.0 


5.7 


18 Sep 36 


2.0 


2.4 


25 Jan 36 
4 Dec 42 
8 Feb 59 



3-89 



Table 3-8. Highest and Lowest Water Levels — Continued 









Highest Water Levels 


Lowest Water Levels 








Above Mean High Water 




Jelow Low Water 




Observation 


Mean 




(feet) 




(feet) 




Location 


Average 


Extreme 


Date of 


Average 


Extreme 


Date of 




Period 


Range 

(feet) 


Yearly 
Highest 


High 


Record 


Yearly 
Lowest 


Low 


Record 


ATLANTIC COAST 


















Norfolk, Va. 


















Sewells Point 


1928-70 


2.5 


2.8 


6.0 


23 Aug 33 


2.1 


3.0 


23 Jan 28 
26 Jan 28 


Morehead City, N.C. 


1953-57 


2.8 





4.2 


15 Oct 54 
19 Sep 55 





1.7 


11 Dec 54 


Wilmington, N.C. 


1935-70 


3.6 


1.7 


4.3 


15 Oct 54 


1.4 


2.0 


3 Feb 40 


Southport, N.C. 


1933-53 


4.1 


2.4 


3.4 


2 Nov 47 


1.2 


1.9 


28 Jan 34 


Charleston, S.C. 


1922-70 


5.2 


2.2 


5.2 


11 Aug 40 


2.4 


3.6 


30 Nov 63 


Fort Pulaski, Ga. 


1936-70 


6.9 


2.5 


4.2 


15 Oct 47 


2.9 


4.4 


20 Mar 36 


Femandina, Fla. 


1897-1924 
1939-1970 


6.0 


2.5 


7.7 


2 Oct 1898 


2.5 


3.9 


24 Jan 1940 


Mayport, Fla. 


1928-70 


4.5 


1.9 


3.0 


9 Sep 64 


2.0 


3.2 


24 Jan 40 


Miami Beach, Fla. 


1931-51 
1955-70 


2.5 


1.6 


3.9 


8 Sep 65 


1.3 


1.6 


24 Mar 36 


GULF COAST 


















Key West, Fla. 


1926-70 


1.3 


1.3 


2.5 


8 Sep 65 


1.2 


1.6 


19 Feb 28 


St. Petersburg, Fla. 


1947-70 


2.3* 


1.6 


3.0 


5 Sep 50 


1.7 


2.1 


3 Jan 58 


Cedar Key, Fla. 


1914-25 
1939-70 


2.6 


2.2 


3.2 


15 Feb 53 
11 Sep 64 


2.8 


3.5 


27 Aug 49 

21 Oct 52 

4 Feb 63 


Pensacola, Fla. 


1923-70 


1.3* 


1.6 


7.6 


20 Sep 26 


1.4 


2.2 


6 Jan 24 


Grand Isle, La. 


















Humble Platform 


1949-70 


1.4* 


1.5 


2.2 


25 Sep 53 
22 Sep 56 


1.4 


1.7 


22 Nov 49 
3 Feb 51 


Bayou Rigaud 


1947-70 


1.0* 


1.9 


3.9 


24 Sep 56 


1.1 


1.5 


3 Feb 51 
13Jan 64 


Eugene Island, La. 


1939-70 


1.9* 


2.5 


6.0 


27Jun 57 


1.6 


2.4 


25 Jan 40 


Galveston, Tex. 


1908-70 


1.4* 


2.2 


10.1 


16,17 Aug 15 


2.6 


5.3 


11 Jan 08 


Port Isabel, Tex. 


1944-70 


1.3* 


1.5 


3.8 


11 Sep 61 


1.4 


1.7 


31 Dec 56 
7 Jan 62 


* Diurnal Range 







3-90 



Table 


3-8. Highest an 


d Lowest Water Levels — Continue 


d 






Observation 
Period 


piurnal 
Range 

(feet) 


Highest Water Levels 

Above Mean High Water 

(feet) 


Lowest Water Levels 

Below Low Water 

(feet) 


Location 


Average 
Yearly 
Highest 


Extreme 
High 


Date of 
Record 


Average 
Yearly 
Lowest 


Extreme 
Low 


Date of 
Record 


PACIFIC COASTt 
San Diego, Calif. 


1906-70 


5.7 


1.8 


2.6 


20 Dec 68 
29 Dec 59 


2.0 


2.8 


17 Dec 33 
17 Dec 37 


LaJolla,CaUf. 


1925-53 
1956-70 


5.2 


1.8 


2.4 


20 Dec 68 


1.9 


2.6 


17 Dec 33 


Los Angeles, Cahf. 


1924-70 


5.4 


1.8 


2.3 


19,20 Dec 68 


1.8 


2.6 


26 Dec 32 
17 Dec 33 


Santa Monica, Calif. 


1933-70 


5.4 


1.8 


2.1 


27 Dec 36 
17,18 Jul 51 


1.8 


2.7 


17 Dec 33 


San Francisco, Calif. 


1898-1970 


5.7 


1.5 


2.3 


24 Dec 40 

19 Jan 69 

9 Jan 70 


2.1 


2.7 


26 Dec 32 
17 Dec 33 


Crescent City, CaUf. 


1933-46 
1950-70 


6.9 


2.4 


3.1 


4 Feb 58 


2.4 


2.9 


22 May 55 


Astoria, Oreg. 


1925-70 


8.3 


2.6 


3.8 


17 Dec 33 


1.9 


2.8 


16 Jan 30 


Neah Bay, Wash. 


1935-70 


8.2 


2.9 


4.1 


30 Nov 51 


3.1 


3.8 


29 Nov 36 


Seattle, Wash. 


1899-1970 


11.3 


2.2 


3.3 


6 Feb 04 
5 Dec 67 


3.9 


4.7 


4Jan 16 
20Jun 51 


Friday Harbor, Wash. 


1934-70 


7.7 


2.3 


3.2 


30 Dec 52 


3.2 


4.0 


7 Jan 47 


Ketchikan, Alaska 


1919-70 


15.3 


4.4 


5.9 


2 Dec 67 


4.5 


5.2 


8 Dec 19 
16 Jan 57 
30 Dec 59 


Juneau, Alaska 


1936-41 
1944-70 


16.4 


5.0 


6.3 


2 Nov 48 


4.5 


5.7 


16 Jan 57 


Skagway, Alaska 


1945-62 


16.7 


4.9 


6.7 


22 Oct 45 


5.0 


6.0 


16 Jan 57 


Sitka, Alaska 


1938-70 


9.9 


3.4 


4.7 


2 Nov 48 


3.2 


3.8 


19Jun 51 
16 Jan 57 


Yakutat, Alaska 


1940-70 


10.1 


3.7 


4.8 


2 Nov 48 


3.1 


3.9 


29 Dec 51 
16Jan 57 


Seward, Alaska 


1925-38 


10.6 


3.4 


4.1 


13 Oct 27 


3.5 


4.2 


14 Jan 30 


Kodiak Island, Alaska 
Womens Bay 


1949-63 


8.8 


2.8 


3.7 


21 Nov 49 


2.9 


3.6 


15,16 Jan 57 


Unalaska Island, Alaska 
Dutch Harbor 


1934-39 
1946-70 


3.7 


2.0 


2.9 


14,15 Jan 38 


2.0 


2.7 


13 Nov 50 


Adak Island, Alaska 
Sweeper Cove 


1943-70 


3.7 


1.8 


2.7 


28 Dec 66 


2.3 


3.3 


11 Nov 50 


Attu Island, Alaska 
Massacre Bay 


1943-69 


3.3 


1.6 


2.4 


6 Jan 66 


1.7 


2.4 


12,13 Nov 50 



t Tsunami levels not included. 



3-91 



1969, in excess of 20 feet MSL over many miles of the open Gulf Coast, 
with a peak value of 24 feet MSL near Pass Christian, Mississippi. High 
water levels in excess of 12 feet MSL on the open coast and 20 feet within 
bays were recorded along the Texas coast as the result of Hurricane Carla, 
September, 1961. Water levels above 13 feet MSL were recorded in the 
Florida Keys during Hurricane Donna, 1960. 

Accumulation of data over many years in some areas, such as regions 
near the North Sea, has led to relatively accurate empirical techniques 
of storm surge prediction for some locations. However, these empirical 
methods are not applicable to other locations. In general, not enough 
storm surge observations are available in the United States to make accu- 
rate predictions of storm surge. Therefore, it has been general practice 
to use hypothetical design storms, and to estimate the storm-induced surge 
by physical or mathematical models. Mathematical models are usually used 
for predicting storm surge, since it is difficult to represent some of the 
storm surge generating processes (such as the direct wind effects and 
Coriolis effects) in physical laboratory models. 

a. Hydrodynamic Equations . Equations that describe the storm surge 
generation processes are the continuity equation expressing conservation 
of mass and the equations of motion expressing Newton's second law. The 
derivations are not presented here; references are cited below. The equa- 
tions of motion and continuity given here represent a simplification of 
the more complete equations. A more simplified form is obtained by verti- 
cally integrating all governing equations and then expressing everything 
in terms of either the mean horizontal current velocity or volume trans- 
port. Vertically integrated equations are generally preferred in storm- 
surge calculations since interest is centered in the free surface motion 
and mean horizontal flow. Integration of the equations for the storm 
surge problem are given by Haurwitz (1951), Welander (1961), Fortak (1962), 
Platzman (1963), Reid (1964), and Harris (1967). 

The equations given here are obtained by assuming: 

(1) vertical accelerations are negligible, 

(2) curvature of the earth and effects of surface 

waves can be ignored, 

(3) the fluid is inviscid, and 

(4) the bottom is fixed and impermeable. 

The notation and the coordinate scheme employed are shown schematic- 
ally in Figure 3-43. D is the total water depth at time t, and is 
given by D = d + S, where d is the undisturbed water depth and S is 
the height of the free surface above or below the undisturbed depth re- 
sulting from the surge. The Cartesian coordinate axes, x and y, are in 
the horizontal plane at the undisturbed water level and the z axis is 
directed positively upward. The x axis is taken normal to the shoreline 
(positive in the shoreward direction), and the y axis is taken alongshore 
(positive to the left when facing the shoreline from the sea). 

3-92 




W77777777777777777777777777Z777777777;77777777Z777Z 

PROFILE 



JiL -^ 



LAND 



^ JZ. 



SL J^ 



■ SHORELINE- 




JU>.A^ .AJU^ >t/OC/ 



SEA 



PLAN VIEW 



Figure 3-43. Notation and Reference Frame 



3-93 



The differential equations appropriate for tropical or extratropical 
storm surge problems on the open coast and in enclosed and semienclosed 
basins are as follows; 

H; »M^, ^ »M^ ^ as ^ a , ,„ il , II' _ :^ , W,P (3-50) 

9t ax 3y ^ 3x ^ ax ^ 3x p p ^ 

o s 

c a 

■C « 
" c 



'■ 


u 


■ 






a, 






.!2 


o 


(U 


1) 


"3 


C/3 


it' 




Ut 


u 


> 







u 




c 




3 




CQ 




C/5 







OJ 




(A 


CO 


4.* 




"c3 


U 


4-> 




4-> 




C/5 




6 

4-* 

< 


c 


E 

o 

o 

pa 


3 













av 9M 3M as aj af 'sy 'by 

^ + -7^+.=~^"g°;- + g°r + g°^'"n"„^%P t3-51) 
at ay ax ay 9y ay p p > 



as 3U av 

— + — + — = P . (3-52) 

at dx ay 

where 

S S S 

M^^ = / uMz; M^^ = / vMz; M^^ = / uvdz; 
-d -d -d 



S S 

/dz 
d -d 



U = / udz; V = / vc 



The symbols are defined as: 

U, V = X and y components, respectively, of the volume 
transport per unit width; 

t = time; 

Mj,^, Myy, M^j^ = momentum transport quantities; 
f = 2(jj sin (|) = Coriolis parameter; 



3-94 



Hi = angular velocity of earth 

(7.29 X 10"^ radians/second); 

<f) = geographical latitude; 

Tg^, Tgj, = X and y components of surface wind stress; 

"^bx' "^bii ~ ^ ^"^ y components of bottom stress; 

p = mass density of water; 

W^, Wj^ = X and y components of wind speed; 

5 = atmospheric pressure deficit in head of water; 

I, = astronomical tide potential in head of water; 

u, V = X and y components, respectively, of current 
velocity; 

P = precipitation rate (depth/time) ; 

g = gravitational acceleration; and 

9 = angle of wind measured counterclockwise from 
the X axis. 

Equations 3-50 and 3-51 are approximate expressions for the equations 
of motion and Equation 3-52 is the continuity relation for a fluid of 
constant density. These basic equations provide, for all practical pur- 
poses, a complete description of the water motions associated with nearly 
horizontal flows such as the storm surge problem. Since these equations 
satisfactorily describe the phenomenon involved, a nearly exact solution 
can only be obtained by using these relations in complete form. 

It is possible to obtain useful approximations by ignoring some terms 
in the basic equations when they are either equivalent to zero or are 
negligible, but accurate solutions can be achieved only by retaining the 
full two-dimensional characteristics of the surge problem. Various sim- 
plifications (discussed later) can be made by ignoring some of the physi- 
cal processes. These simplifications may provide a satisfactory estimate, 
but they must always be considered as only an approximation. 

In the past, simplified methods were used extensively to evaluate 
storm surge because it was necessary to make all computations manually. 
Manual solutions of the complete basic equations in two dimensions were 
prohibitively expensive because of the enormous computational effort. 
With high speed computers, it is possible to resolve the basic hydro- 
dynamic relations efficiently and economically. As a result of computers, 
several workers have recently developed useful mathematical models for 
computing storm surge. These models have substantially improved accuracy, 
and provide a means for evaluating the surge in the two horizontal dimen- 
sions. These more accurate methods are not covered here, but are highly 
recommended for resolving storm-surge problems where more exactness is 



3-95 



warranted by the size or importance of the problem. These methods are 
recommended only if a computer is available. A brief description of 
these methods and references to them follows. 

Solutions to the basic equations given can be obtained by the tech- 
niques of numerical integration. The differential equations are approxi- 
mated by finite differences resulting in a set of equations referred to 
as the numerical analogs. The finite-difference analogs, together with 
known input data and properly specified boundary conditions, allow evalua- 
tion at discrete points in space of both the fields of transport and water 
level elevations. Because the equations involve a transient problem, 
steps in time are necessary; the time interval required for these steps is 
restricted to a value between a few seconds and a few minutes depending on 
the resolution desired and the maximum total water depth. Thus solutions 
are obtained by a repetitive process where transport values and water-level 
elevations are evaluated at all prescribed spatial positions for each time 
level throughout the temporal range. 

These techniques have been applied to the study of long-wave propa- 
gation in various water bodies by numerous investigators. Some investi- 
gations of this type are listed below. Mungall and Matthews (1970) devel- 
oped a variable-boundary, numerical tidal model for a fjord inlet. The 
problem of surge on the open coast has been treated by Miyazaki (1963), 
Leendertse (1967), and Jelesnianski (1966, 1967, and 1970). Platzman 
(1958) developed a model for computing the surge on Lake Michigan result- 
ing from a moving pressure front, and also developed a dynamical wind tide 
model for Lake Erie. (Platzman, 1963.) Reid and Bodine (1968) developed 
a niimerical model for computing surges in a bay system taking into account 
flooding of adjacent low lying terrain and overtopping of low barrier 
islands. 

b. Simplified Techniques for Determining Storm Surge . The tech- 
niques described here for the determination of storm surge are simple, and 
it is possible to carry out all storm surge calculations manually, using a 
desk calculator or slide rule. In most cases, however, it is desirable to 
employ a digital computer for the computations to reduce the effort and to 
improve accuracy. It is sometimes possible to estimate surge with satis- 
factory accuracy using a set of simplified equations, if the particular 
problem is not too complex, and if the simplified technique can be verified 
from actual prototype field data. Simpler schemes for computing storm 
surge are obtained by including only those phenomena that appear signifi- 
cant to the investigation; thus some of the less important terms are 
omitted from Equations 3-50, 3-51 and 3-52. 

(1) Storm Surge on the Open Coast . Ocean basins are large and 
deep beyond the shallow waters of the Continental Shelf. The expanse of 
ocean basins permits large tropical or extratropical storms to be situated 
entirely over water areas allowing tremendous energy to be transferred 
from the atmosphere to the water. Wind-induced surface currents, when 
moving from the deep ocean to the coast, are impeded by the shoaling 
bottom, causing an increase in water level over the Continental Shelf. 



3-96 



Onshore winds, cause the water level to begin to rise at the edge of the 
Continental Shelf. The amount of rise increases shoreward to a maximum 
level at the shoreline. Storm surge at the shoreline can occur over long 
distances along the coast. The breadth and width of the surge will depend 
on the storm's size, intensity, track and speed of forward motion as well ' 
as the shape of the coastline, and the offshore bathymetry. The highest 
water level reached at a location along the coast during the passage of a 
storm is called the maximum surge for that location; the highest maximum 
surge is called the peak surge. Maximum water levels along a coast do not 
necessarily occur at the same time. The time of the maximum surge at one 
location may differ by several hours from the maximum surge at another 
location. The variation of maximum surge values and their time for many 
locations along the east coast during Hurricane Carol, 1954, are shown in 
Figure 3-44. This hurricane moved a long distance along the coast before 
making landfall, and altered the water levels along the entire east coast. 
The location of the peak surge relative to the location of the landfall 
where the eye crosses the shoreline depends on the seabed bathymetry, wind- 
field, configuration of the coastline, and the path the storm takes over 
the shelf. For hurricanes moving more or less perpendicular to a coast 
with relatively straight bottom contours, the peak surge will occur close 
to the point where the region of maximum winds intersects the shoreline, 
approximately at a distance R, to the right of the storm center. Peak 
surge is generally used by coastal engineers to establish design water 
levels at a site. 

Attempts to evaluate storm surge on the open coast, and also in bays 
and estuaries, when obtained entirely from theoretical considerations, 
require verification particularly when simplified models are used. The 
surge is verified by comparing the theoretical system response and computed 
water levels with those observed during an actual storm. The comparison is 
not always simple, because of the lack of field data. Most water-level 
data obtained from past hurricanes were taken from high water marks in low- 
lying areas some distance inland from the open coast. The few water-level 
recording stations along the open coast are too widely separated for satis- 
factory verification. Estimates of the water level on the open coast from 
levels observed at inland locations are unreliable, since corrective adjust- 
ments must be made to the data, and the transformation is difficult. 

Systematic acquisition of hurricane data by a number of organizations 
and individuals began during the first quarter of this century. Harris 
(1963b) presented water-level data and synoptic weather charts for 28 
hurricanes occurring from 1926 to 1961. Such data are useful for verifying 
surge prediction techniques. 

Because of the limited availability of observed hurricane surge data 
for the open coast, design analysis for coastal structures is not always 
based on observed water levels. Consequently a statistical approach has 
evolved that takes into account the expected probability of the occurrence 
of a hurricane with a specific CPI at any particular coastal location. 
Statistical evaluation of hurricane parameters based on detailed analysis 
of many hurricanes, have been compiled for coastal zones along the Atlantic 
and Gulf coasts of the U.S. The parameters evaluated were the radius of 

3-97 



90" 



80<» 



— 40<> 




NewLondorL, 
Willets Point 
Batteryj; 



I Sandy Hook 

I Philadelphia* 



70° 



'^•B'ar Hartior 
'"Portland 
'Portsmouth 
•"^Boston 

Woods Hole 
Newport 
Montauk 
31 




Baltimore* 
Annapolis* 
Solomon ' 



Old Point Comfort^ 
Hampton Roads! 
Portsmouth^ 



Morehead Citv 
Wilmington, 
Southport 



+ 



Atlantic City 
Breakwater Harbor 



/ 



[Little Creek 



Charleston. 
Savannah*. 



/ 



30 



29 



./ 



;302 



+ 



S" 



27 



+ 



Key: 

Development Stage •••••• 

Tropical Stoge ^^^i 

Hurricane Stoge •■^-^ 

Frontal Stage «»*^» 

O Position at 0700 EST 

• Position at 1900 EST 



90" 

I 




70° 



40° — 



30° — 



(from Harris, 1963 b) 



Figure 3-44. Storm Surge Chart. Hurricane Carol, 30, 31 August 1951, 
Insert Map for New York Harbor 



3-98 




Aug 30 



Aug 31 



Aug 28 Aug. 29 Aug 30 Aug. 31 



SepI 



Figure 3-44. Storm Surge Chart. Hurricane Carol, 30, 31 August 1951. 
Insert Map for New York Harbor -- Continued 



3-99 



maximum wind R; the minimiom central pressure of the hurricanes p ; 
the forward speed of the hurricane V„, while approaching or crossing 

the coast; and the maximum sustained wind speed W, 30 feet above the 
mean water level. 

Based on this analysis, the U.S. Weather Bureau (now the National 
Weather Service) and U.S. Army Corps of Engineers jointly established 
specific storm characteristics for use in the design of coastal struc- 
tures. Because the parameters characterizing these storms are specified 
from statistical considerations and not from observations, the storms 
are termed hypothetical huvvioanes or hypo-hurriaanes . The parameters 
for such storms are assumed constant during the entire surge generation 
period. Graham and Nunn (1959) have developed criteria for a design storm 
where the central pressure has an occurrence probability of once in 100 
years. This storm is referred to as the standard project hurr-ioane (SPH) . 
The mathematical model used for predicting the wind and pressure fields in 
the SPH is discussed in Section 3.72, Model Wind and Pressure Fields for 
Hurricanes. The SPH is defined as a "hypo-hurricane that is intended to 
represent the most severe combination of hurricane parameters that is 
reasonably oharaateristia of a region excluding extremely rare combina- 
tions." Most coastal structures built by the U.S. Army Corps of Engineers 
that are designed to withstand or protect against hurricanes are based on 
design water associated with the SPH. 

The construction of nuclear-powered electric generating stations in 
the coastal zone made necessary the definition of an extreme hurricane 
called the probable maximum hurricane (PMH) . The PMH has been adopted by 
the Atomic Energy Commission for design purposes to ensure public safety 
and the safety of nuclear-power facilities. Procedures for developing a 
PMH for a specific geographical location are given in U.S. Weather Bureau 
Interim Report HUR 7-97 (1968). The PMH was defined as "A hypothetical 
hurricane having that combination of characteristics which will make the 
most severe storm that is reasonably possible in the region involved, if 
the hurricane should approach the point under study along a critical path 
and at an optimum rate of movement." 

Selection of hurricane parameters and the methods used for developing 
overwater wind speeds and directions for various coastal zones of the 
United States are discussed in detail by Graham and Nunn (1959) and in 
HUR 7-97 (1968) for the SPH and PMH. The basic design storm data should 
be carefully determined, since errors may significantly affect the final 
results. 

Two simple methods are presented here for estimating storm surge on 
the open coast: one a quasi-static numerical scheme and the other a nomo- 
graph method. These methods should never be used for estimating the surge 
in bays, estuaries, rivers or in low-lying regions landward of the normal 
open-coast shoreline. Neither method is entirely satisfactory for all 
cases, but for many problems both appear to give reasonable solutions. 
The use of each method is illustrated by estimating the peak open- coast 
storm surge for an actual hurricane. The peak surges thus calculated are 
compared to the surge computed by a complete two-dimensional numerical 
model for the same storm. 

3-100 



(a) Quasi-Static Method for Prediction of Hurricane Surge . 
TTiis method for determining open-coast storm surge is based on theoretical 
approximations of the governing hydrodynamic equations originally proposed 
by Freeman, Baer, and Jung (1957). The term quasi-static method is used 
here to emphasize that this method should be restricted to slow-moving, 
large-scale storm systems. This method is called the Bathys trophic Storm 
Tide Theory and, unlike earlier one-dimensional theories, some of the 
effects of longshore flow and the apparent Coriolis force are considered. 
Such an approximation of the theory can be described as a quasi-static 
method in which a numerical solution is obtained by successively integrat- 
ing wind stresses over the Continental Shelf from its seaward edge to the 
shore for a predetermined interval of time. 

This simplified method assumes that storm surge responds instanta- 
neously to the onshore wind stresses, advection of momentum can be ignored, 
longshore sea surface is uniform, and no flow is assumed normal to the 
shore which is treated as a seawall. Barometric effects and precipita- 
tion also can be neglected. Setup due to atmospheric pressure difference 
can be estimated from another source, and added to the final design water 
level. Based on the preceding assumptions. Equations 3-50 and 3-51 reduce 
to 

gD — = fV + -^ (3-53) 



3x 

ay _ ^sy - '^by 
3t p 



(3-54) 



Conservation of mass is not considered because, (1) there is no flow 
perpendicular to shore, (2) the longshore flow is assumed independent of 
y and, (3) the water level is assumed slowly changing. The forces (ex- 
pressed in mass times acceleration per unit area) involved and the corres- 
ponding response of the sea for the bathystrophic approximation are shown 
in Figure 3-45. As indicated in the figure, the surface shear force act- 
ing in the x-direction Xg^ and the apparent Coriolis force is balanced 
by the hydrostatic pressure force pgA(9S/3x). Moreover, the surface shear 
force acting in the y-direction t is balanced by the bottom shear 
force T^jy and the inertial force p(9V/9t). 

Bretschneider and Collins (1963) developed a computational model 
based on this Bathystrophic Theory and applied it to open- coast surge 
problems for the region around Corpus Christi, Texas. Marinos and 
Woodward (1968) modified the Bretschneider and Collins model and calibra- 
ted it for various reaches along the Texas coast by using three hurricanes 
of record, and also made parametric studies of hypo-hurricane surges for 
the entire coast of Texas. 

In some cases the underlying assumptions made in the development of 
this theory are not satisfied. Thus, as a consequence of assuming that 
the onshore wind stresses cause an instantaneous change of the water level, 
i.e., U = 0, the traverse line (the line over the Continental Shelf along 
which computations are carried out) must always be taken perpendicular to 

3-101 



» o 
A — 



> o 



o ° 



■D _ 

m » O 

2 »- "- 

O o o 



> TJ Q. 



o 

z 




2 § 

II n 



o 

•H 
4-> 

§ 

X 
o 
(^ 

u 

•H 

o 
■p 

M 
>s 

'P 

cd 

CQ 

^1 

.2 

(A 

in 

n 
o 
p< 
in 

iJ 

g 

tn 
Q> 
u 
^ 
o 
u, 

(4-1 
O 



0) 

u 



I 
V 

•H 

[L, 



I- ». Q- o« ^ 



3-102 



the bottom contours for valid computations. The above assumption also 
implies that there is no flooding landward of the shore; thus, there is 
a deficiency in the method when substantial flooding occurs. 

The bottom contours of the actual seabed are rarely straight and para- 
llel; however, the traverse line can often be oriented so that it is nearly 
perpendicular to the contours in an average way. For complex offshore 
bathymetry, such an approximation would be invalid. For storms moving 
more or less perpendicular to a coastline, the traverse line can be taken 
through, or anywhere to the right of, the region of maximum winds, but 
never to the left of this region. Many other factors such as the angle 
of approach of a storm, the coastline configuration, and inertial effects, 
limit the use of such a simple approach. 

The computation model given here is based on Bathystrophic Storm Tide 
Theory as described by Bodine (1971). Although Bodine applied both manual 
and digital computer calculation methods to the open coast storm surge 
problem, only the manual method is presented. 

The bottom and surface shear stresses are assumed to vary according 
to: 



^by KVIVI 



sx 
^- = kW^ cos e 



T 

~p 

T 



(bottom shear stress) (3-55) 



(wind shear stress) (3-56) 



^ = kW^ sin e 



in which K is a dimensionless bottom friction coefficient, k is a 
dimensionless surface friction coefficient, W is the wind speed, and 
6 is the angle between the x-axis and the local wind vector. The bottom 
friction coefficient K is related to the coefficient of Chezy C and 
the Darcy-Weisbach friction factor f^ as follows: 

Typical bottom conditions result in a value of K that lies in the 
range between 2 x 10"^ and 5 x 10" ■^. For a first estimate, a value 
of K = 2.5 x 10"^ may be assumed. This coefficient is used in cali- 
brating the model. It not only accounts for energy dissipation at 
the bed, but may be used to adjust for inexact modeling and deficien- 
cies caused by ignoring some of the hydrodynamic processes involved. 



3-103 



The wind stress coefficient is based on one given by Van Dom (1953) and 
others which is assiomed to be a function of wind speed; thus, 

k = X , for W < W^ (3-58) 




as 
ax 


= ^ [fv+ kw^cose] 


9V 
3t 


= kW^ sin - ^^^ 
D 



(3-59) 

where the constants Kj^ and K2 are usually taken to be 1.1 x 10"^ and 
2.5 X 10"^, respectively. W^ is a critical wind speed taken as 14 knots 
(16 miles per hour) . 

Introducing the bottom and free-surface stress relations into 
Equations 3-53 and 3-54 gives 

(3-60) 



(3-61) 



an estimate of the amount of setup that can be attributed to the onshore 
effects. The setup attributed to longshore effects can be obtained by 
rewriting Equation 3-60 in the following two-component form: 

^ _ kW^ cos 6 (3-62) 

ax " gD 

^ = iY^ (3-63) 

ax gD 

The total setup along the x-axis is the sum of the two components or, 

^ = !!^ + !!z (3-64) 

ax ax ax 

In the finite-difference niomerical solution of the reduced equations, 
values of S^ and Sj. are evaluated at points spaced Ax apart along a 
single Cartesian axis (the traverse line). The values of S, the total 
setup for the increment, can be regarded as being the water level midway 
between two points along the traverse line, midway between x and x + Ax 
at a time t. The longshore volume transport of water V is also evaluatei 
between points. Wind stress data and the Coriolis parameter are supplied 
at the points x and x + Ax. The subscripts and superscripts i and n 

3-104 



are used to denote discrete points in space and time, respectively. The 
quantities Ax and At are allowed to be nonuniform spacings along the 
traverse line and time increments, respectively. The finite difference 
forms of Equations 3-61, 3-62, and 3-63 are then given as follows: 



Ax 






n + l 



(As,)r.v! 



^^^- If +f \ v"""^ 






1 



B.- + B.>i)" + (B.- + B.^,) 



n + l 



At + V 



+ 'A 



l + K|V«,^JAt(D-)-^^ 



(3-65) 
(3-66) 

(3-67) 



where A and B are the kinematic forms of the wind stress given by: 



A = k W cos 



B == k W^ sin 



(3-68) 
(3-69) 



The time step specified by the ordinal number n represents a time 

level at which AS^, AS^, and V are known, while n+l represents the 

new time level for which the quantities AS^, ^Sy and V are to be 
determined. 

The total water depth at the mid-interval between two time steps is 
given by: 



i + Js 2 ^ 



„n „n + l 

^A ■*■ ^A 



+ (s. + s),;,^ 



<^ 



(3-70) 



(%),• + {^Ap)i + l]" + [(%).■ + {^Ap)i + lY^'] 



where 



Sg = initial setup 



S^ = astronomical tide 



S^p — atmospheric pressure setup. 



3-105 



The initial setup refers to the water level at the time the storm surge 
computations are started. An approximate relationship giving the atmos- 
pheric pressure setup in feet when pressure is expressed in inches of 
mercury is 



Sap = 1-14 (p„-pj d-e-'^) 



(3-71) 



where p„ is the pressure at the periphery of the storm, and r is the 

radial distance from the storm center to the computation point on the 
traverse line. 

The total depth at the new time level is evaluated by the relation: 



D 



n + 1 

+ 'A 



d,.+ d,.^, 



+ s, + s^"^+ (s. + sj;,, 



'A 



(3-72) 



(Sap), + (s^p),.,] 



n + l 



The total water level rise at the coast is the sum of all component water 
level changes resulting from the meteorological storm plus those components 
which are not related to the storm. Hence the total setup is given by 



s. + S3, + s^p + s, + S^ + S^ + S^ 



(3-73) 



where Sj/ is a component due to the wave setup in the region shoreward of 
the breaker line and is related to the breaker height by Equation 3-49. 
(See Section 3.85, Wave Setup.) The setup component S^ accounts for the 
setup resulting from local conditions such as bottom configuration, coast- 
line shape, or other flows influencing the system, such as flows from bay 
inlets or rivers. This component can only be estimated from a full under- 
standing of the influence of topographic and hydrographic features not 
considered in the numerical calculations. Contributions made by the 
various setup components to the total surge are shown in Figure 3-46. 

Equations 3-65, 3-66, and 3-67 can be rewritten in a more useful form 
by introducing dimensional constants for terms frequently appearing in the 
equations. Hence, 



(As.)r;4 



C, Ax 



1 Ax 
A. 



+ ^.l) 



n+l 



(3-74) 



i+'A 



(-^)::i 



C2 Ax 



D' 



n + l 
i+'A 



(sm0),. + (sin0).^^] V^^,)^ 



(3-75) 



n + l 
^i+'A 



(B, + B.,,)«+(B, + B.,,)«-^](At)+ V",, 

l+C3lV«,^JAtK(D-)r;^^ 
3-106 



(3-76) 



22 
20 

18 

E 

^ 16 

o 
O 

w 14 

z 

5 12 

o 



o 

o 
> 



a. 
a> 
O 

E 
o 

o 
m 



100 
200 
300 
400 
500 



600 



\ / 


^^ Design Wafer Level Including 
Surface- Wove Sefup 
























r 


■ — IJ 




__ 


_Sw 


\ 






M 










\ 






s 


X 


\ 








\ 










\ 


\, 






^ 






— _ 




\ 


s 




\ 






\ 




Sy^ 

-4- 




--..^ 


\ 






\ 








~- 




^^ 




\ 
















\ 


















\ 




S 

1 


\ 
A 












\ 




] 












(s. 


>-^ 


Meon 


Seo Level 
enfol Sh< 


(MSL) 






>v 






— Contin 


If 








LEG 


END 




^^--^ 


\ 






S^ - 

c 


Breaking Wave Setup 
X -Component Setup 
y- Component Sefup 
Atmospheric Pressure Setup 
Asfronomicol Tide 




\ 






1 s, . 
Sy ■ 










i'Ap 










Se 


: In 


tiol VI 


*ater Leve 


1 







400 200 

Distance in Yards 



60 



10 20 30 40 50 

Distance from Coast (Nautical Miles) 

(from Bodine,l97l ) 

Figure 3-46. Various Setup Components Over the Continental Shelf 



3-107 



The values of the dimensional constants Cj, C2, and C3 in Equations 
3-74, 3-75 and 3-76 depend on the units used in performing the calculations. 
Table 3-9 gives the dimensions of the variables used in the numerical 
scheme in three systems of units and the corresponding value of the 
constants for each system. The first column of units is given in the 
metric system while the other two are given in mixed units in the English 
system. 

Table 3-9. Systems of Units for Storm Surge Computations 



Parameters 


Units and Constant Values 


Metric 


Mixed English 


Ax 
AS^, ASy 

g 
D 

A, B 

V 

f 

At 

Cl 

C2 

C3 


km 

m 

m/sec^ 

m 
(km/hr) 2 

km2/hr 

hr-l 

hr 

3.94 

2.06 
(1000)2 


nm 

ft 

ft/sec2 

ft 
(nm/hr) 2 

nm2/hr 

hr-1 

hr 

269 

141 
(6080)2 


mi 
ft 

ft/sec2 

ft 
(mi/hr) 2 

mi2/hr 

hr-l 

hr 

176 
92 
(5280)2 



The assumptions for this model are reasonably good only when the 
momentum of the water is increasing under the influence of the wind, i.e, 
when the right side of Equation 3-76 is positive. This condition can be 
considered in the calculations by using the relation 



^n + 1 ^ 




+ B, 



\n+l 



i+1 



(^%i 



2C, K 



(3-77) 



For a derivation of this equation see Bodine (1971). Should V at a 
particular time level, as evaluated from Equation 3-76, be less than the 
value obtained from Equation 3-77, then the result from Equation 3-77 is 
ignored. However, if the opposite is true, then as an estimate, V is 
taken equal to the maximiom possible value as given by Equation 3-77. 



3-108 



It is customary to assume that the system is initially in a state of 
equilibrium and V = 0, and S is uniform along the traverse line at the 
start of the computational scheme. Thus, computations should be initiated 
for conditions prior to the arrival of strong winds over the Continental 
Shelf. Although the real system would seldom, if ever, be in a complete 
initial state of equilibrium, errors in assuming it to be are of little 
consequence in the computational scheme after several time steps in the 
calculations, because the effects of the forcing functions will eventually 
predominate . 

To demonstrate the computational procedures, the storm surge in the 
Gulf of Mexico resulting from Hurricane Camille (1969) is calculated. 
Hurricane Camille was an extremely severe storm that crossed the eastern 
part of the Gulf of Mexico with the eye of the storm making landfall at 
Bay St. Louis, Mississippi, at about 0500 Greenwich Mean Time (GMT) on 
18 August 1969. Unusually high water levels were experienced along the 
gulf coast during the passage of Camille because of the intense winds and 
relatively shallow water depths which extend far offshore. The storm surge 
is calculated and compared with the peak surge generated by Camille. 

Information published by the U.S. Weather Bureau for Hurricane Camille 
in HUR 7-113 (1969) indicates that R = 14 nautical miles and Wp = 13 knots 
would be representative of these values which are assumed to be invariant 
while the storm moved over the Continental Shelf. The wind data and track 
for Hurricane Camille have been published by the Weather Bureau in HUR 
7-113A (1970). The track, together with the traverse line used in the 
present calculations, is shown in Figure 3-47. Overwater wind speeds and 
directions are shown in Figure 3-48. A profile of the seabed along the 
traverse line is shown in Figure 3-49. 

Tide records from the region affected by the storm show the mean water 
level to be about 1.2 feet above normal before being affected by the storm. 
This value is taken to represent the initial water level. The range of 
astronomical tides in this location is about 1.6 feet. A constant value of 
0.8-foot above MSL is used in the computations; the final surge hydrograph 
at the coast can be subsequently corrected to account for the predicted 
variations of the tide. Variations in the initial water level and astro- 
nomical tides may be added algebraically to the storm surge calculations 
without seriously affecting the final results. The atmospheric pressure 
difference, P„ - P„. is needed for evaluating the pressure setup com- 
ponent, S^p. Data given in HUR 7-113 (1969) suggest that p^ = 26.73 
inches of mercury and p^ = 29.92 inches of mercury are representative for 
the hurricane. All of these values are assumed to remain constant for the 
calculations . 

The wind data, basin profile, and hurricane characteristics provide 
the basic information needed in making an estimate of the peak storm surge 
associated with Hurricane Camille. The time intervals, distance increments 
along the traverse line, etc., are given in the computational steps to 
follow. 



3-109 




3-110 




a 
o 

u 

3 

O, 

3 
ti 



n 



o 

JJ ON 



in 
D 

E 
o 



< 



m 
o 



in 



0) 

% 

to 



(D 


1— 1 


O 


•H 


M-l 




^ 


CJ 


P 




C/3 


a; 




c 


+J 


rt 


O 


o 


O 


•H 


1 


^ 


o 


:S 


00 




•* 




to 




d> 




U 




3, 






3-1 





V.^ ^ i 


_— t-'s 


°' -^ 


* / » 


-^^ 




' 7/ 




^\ 


^ 


^- V' 


V Ifm ^ 


"V^ 


h>^. 


■»■ 






\ 


o 

in 


°Wm 


'^!S^ 


"\\ \ 


^ 


\w{ 




w) 






^^^^^ 




\ 




S ^-^ y7 C, ^ 


V 


/ , 


^ 








9- 


°. J2 






|o=.ir^ '» ^1^7 °^ 


r- 



On 

\o 

3 
< 



o 

o 
o 
o 
o 



u* 



3-112 



1 


_^\ ~ 


S, 




°5 ; . 




s 




'° W^ 


\ 


\ 
^ \ 

\ 


\ 




s 

\ 

\ 
\ 

\ 


\ 


\ 
\ 
\ 

\ 

~ 1 


*?^wn 




\ 


4. 


1 


\ 


^'^1 




1 

1 




•=^ 




1 

n 


' 


i'^i ^t^T- 


i ^' 







a> 

•H 
4J 

c 
o 
u 

I 
I 



u 







o 



> 
o 



o 
o 
rt 

t+H 
(h 
3 

C/D 

•t-) 
O 
O 

I 

o 



00 
I 

to 

•H 



3~II3 




o 



o 






GO 




to 


o 






r^ 




3 
< 

a> 


o 




— 


ID 


^^ 


E 




ay 


n 




0) 


O 




2 


O) 







c 




o 


o 




o 


o 








o 


3 


L. 


lO 


O 


3 
X 




<u 


o 




o 


M— 




CO 


0) 


o 

•si- 


E 

o 


3 






a> 




« 


.' 




u 


H— 




c 


o 




o 


\— 






n 




«> 






n 


■o 


o 




(U 


to 




o 
</) 
oil 


o 




to 


Od 




3 



(MIW «0|9q4aai) q|daa 



3-114 



This quasi-linear computational scheme can he used for manual com- 
putations; however, since the calculations are repetitive, they can be 
performed more efficiently by us,ing a digital computer. When carried out 
manually, the technique is laborious, tedious and subject to error. 
Bodine (1971) gives a computer program based on the numerical scheme 
presented here. Other programs bas«d on similar niomerical schemes using 
the quasi-static methods have been developed. 

When manual computation of storm surge is necessary, a systematic, 
tabular procedure must be adopted to permit stepping through all of the 
discrete computational points in space for each time increment. Table 
3-10 represents a recommended procedure. One table is required for each 
time increment. Table 3-10 corresponds to the time of peak surge for 
Hurricane Camille; preceding tables required to bring the calculations to 
this point are not included since those calculations are similar. The 
first table in the series must reflect the initial conditions; thus V is 
taken to be zero and S is assumed uniform over the system. 

Manual surge calculations for Hurricane Camille give a peak surge of 
25.03 feet, say 25 feet (MLW) or 24.2 feet (MSL) . The bottom friction 
coefficient selected for this particular example was K = 0.003 and the 
surge was found to be insensitive to small changes in the friction coeffi- 
cient. Computer calculations using a friction coefficient of 0.003 result- 
ed in a peak surge of 25.19 feet and a bottom friction coefficient of 
0.0025 resulted in a peak surge of 25.40 feet. For some basins and storm 
systems, the bottom shear stresses are more significant in determining 
water levels. Therefore, it is important to select a bottom friction 
coefficient by verification (i.e., by comparing calculated results with 
observed water levels). After such verification, the model may be used 
to estimate the storm surge from hypothetical hurricanes for the same 
geographical region. 

The surge hydrograph (water level as a function of time) for Hurricane 
Camille is shown in Figure 3-50 for the most landward computational point 
on the traverse line. This figure shows that the water level rose for 
about the first 8 hours, hut then began to fall gradually until about 27 
hours of computational period had elapsed, then began to rise rapidly. 
A study of the local wind fields during this period shows that the winds 
had an onshore component in the early stages of the storm, then the winds 
began blowing offshore for several hours before the principal rise at the 
coast. 

(b) Nomograph Method . A simplified method for obtaining 
a first approximation to the peak storm surge of a hurricane can be based 
on an empirical analysis of past records, an empirical analysis of a sys- 
tematic set of calculations with numerical models, or a combination of the 
two. Jelesnianski (1972) combined empirical data from Harris (1959) with 
his theoretical calculations to produce a set of nomograms that permit the 
rapid estimation of peak surge for any geographical location when a few 
parameters characterizing a storm are known. 

The first nomogram (Fig. 3-51) permits an estimate of the peak surge 
Sj, generated by an idealized hurricane with specified CPI and radius of 

3-115 



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Figure 3-51. Preliminary Estimate of Peak Surge 



3-119 



maximum wind that is moving perpendicular to a shoreline with a speed of 
15 MPH. This nomogram indicates there is a critical storm size as reflec- 
ted by the radius of maximum winds, R. For a given pressure drop greater 
than zero, the highest peak surge is produced for a critical value of R 
equal to 30 miles and any value of R greater or less than this value 
results in a lesser value of the peak surge. 

A second factor F^ given in Figures 3-52 and 3-53 adjusts for the 
effects of variations in bathymetric characteristics along the gulf and 
Atlantic coasts. A third factor Fm given in Figure 3-54, adjusts for 
the effects of storm speed and the angle with which the storm track 
intercepts the coast . 

The predicted peak storm surge Sp is then given by 

Sp = h Fs ^M (3-78) 

Jelesnianski (1972) applied the scheme to the 43 storms given by Harris 
(1959) that entered land south of New England during the period from 1893 
to 1957. The peak surges reported by Harris are plotted against the peak 
surges predicted by the nomograph method in Figure 3-55. The two- 
dimensional hurricane model and storm surge prediction model described by 
Jelesnianski (1967) was used for all calcualtions without adjustment for 
local variations in friction coefficient or other efforts to calibrate 
the model for individual storms. For many of the hurricanes, the post- 
storm surveys conducted were of limited scope and probably did not disclose 
the true peak surge. Thus, at least a part of the spread between observed 
and computed values must be due to the observed data. In addition to the 
peak surge, other nomograms for computing other storm surge parameters are 
given by Jelesnianski (1967). 

An example problem illustrating the use of the nomogram method follows: 
************** EXAMPLE PROBLEM ************** 
GIVEN : Parameters for Hurricane Camille are: 

Ap = 3.19 inches of mercury (in. Hg.) 
Vp = 13 knots 
R = 14 nautical miles (n.m.) 

FIND : Estimate peak open-coast surge produced by Hurricane Camille. 

3-120 



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4 6 8 10 12 

Computed Peak Surge, Sp (Feet above MSL) 



Figure 3-55. Comparison of Observed and Computed Peak Surges 
(for 43 storms with a landfall south of New England 
from 1893-1957) 



3-124 



SOLUTION: Because of the units used in the various nomograms, the units 
of the parameters are converted to 

3.19(1000) 

Ap = 3.19 in. He. = = 108 milhbars . 

^ ^ 29.53 



Note: 29.53 inches of Hg. « lOOO millibars 

Yp = 13 knots * 14.5 mph 
R = 14 n.m. * 15.6 miles. 

Using Figure 3-51 with values of Ap and R, 

Sj «s 22 feet (MSL) 

Based on the approximate landfall location (west of Biloxi) of 
rricane Camillc 
from Figure 3-52, 



Hxorricane Camille (Fig. 3-47), the shoaling factor F is determined 

s 



F^ ^ 1.23 . 

To evaluate the correction for storm motion Fm the angle of storm 
approach to the coast tj; must be first determined. The definition of 
the angle \l) is shown schematically in the insert in Figure 3-54. 
From Figure 3-47 ^ is estimated to be 102°. From Figure 3-54 for 
4) = 102° and Vp = 14.5 mph, 

^M * 0.97 . 
The peak surge given by Equation 3-78 



Sp = (22) (1.23) (0.97) 

Sp = 26.2 feet, say 26 feet (MSL) 



3-125 



Jelesnianski (1972) has calculated the open coast surge for 
Hurricane Camilla with a full two-dimensional mathematical model. The 
maximum surge envelope along the coast, based on computations from this 
model, is shown in Figure 3-56 where the zero distance corresponds to 
the point of landfall of the hurricane eye. This figure shows that 
the peak surge estimated by this method is about 25+ feet MSL. 

Thus, by three independent estimates, it has been found that the peak 
surge is about 25 feet MSL which corresponds approximately to that 
observed (U.S. Geological Survey) at Pass Christian, Mississippi of 
24.2 feet MSL. It would be expected that a slightly higher peak water 
elevation occurred because Pass Christian is located a few miles left of 
the position where the maximum winds made landfall. 

It is rare that such a close agreement is found when estimating the 
peak surge with these dissimilar models. Normally, because of the 
difference in these predictive schemes, it can be expected that peak 
surge estimates may deviate by as much as 25 percent. For well- 
formulated schemes properly applied, there is usually a trade-off 
between reliability of the estimate and the computational effort. 

************************************* 

(c) Predicting Surge for Storms other than Hurricanes . 
Although the basic equations for water motion in response to atmospheric 
stresses are equally valid for nonhurricane tropical and extratropical 
storms, the structures of these storms are not nearly so simple as that 
of a hurricane. Because the storms display much greater variability in 
structure, it is difficult to derive a proper wind field. Moreover, no 
system of classifying these storms by parameters has been developed 
similar to hurricane classification by such parameters as radius to 
maximum winds, forward motion of the storm center, and central pressure. 

Criteria however have been established for a Standard Project North- 
easter for the New England coast north of Cape Cod as given by Peterson 
and Goodyear (1964). Specific standard-project storms other than hurri- 
canes are not presently available for other coastal locations. Estimates 
of design-stoim wind fields can be made by meteorologists working directly 
with climatological weather maps, and by use of statistical wind records 
on land and assuming that they blow toward shore for a significant dura- 
tion over a long, straight line fetch. 

Once the wind field is determined, estimation of the storm surge may 
be determined by methods based on the complete basic formulas or the quasi- 
static method given. The nomogram method cannot be used, since this scheme 
is based on the hurricane parameters. 

(2) Storm Surge in Enclosed Basins . An example of an inclined 
water surface caused by wind shearing stresses over an enclosed body of 
water occurred during passage of the hurricane of 26-27 August 1949 over 
the northern part of Lake Okeechobee, Florida. After the lake level was 
inclined by the wind, the wind direction shifted 180° in 3 hours, resulting 



3-126 



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3-127 



in a turning of the height contours of the lake surface. However, the 
turning of the contours lagged behind the wind so that for a time the 
wind blew parallel to the water level contours instead of perpendicular 
to them. Contour lines of the lake surface from 1800 hours on 26 August 
to 0600 hours on 27 August 1949 are shown in Figure 3-57. The map con- 
tours for 2300 hours on 26 August show the wind blowing parallel to the 
highest contours at two locations. (Haurwitz, 1951), (Saville, 1952), 
(Sibul, 1955), (Tickner, 1957), and (U.S. Army, Corps of Engineers, 1955).) 

Recorded examples of wind setup on the Great Lakes are available from 
the U.S. Lake Survey, National Ocean Survey, and NOAA. These observations 
have been used for the development of theoretical methods for forecasting 
water levels during approaching storms and for the planning and design of 
engineering works. As a result of the need to predict unusually high 
stages on the Great Lakes, numerous theoretical investigations have been 
made of wind setup for that area. (Harris, 1953), (Harris and Angelo, 
1962), (Platzman and Rao, 1963), (Jelesnianski, 1958), (Irish and 
Platzman, 1962), and (Platzman, 1958, 1963, 1965, and 1967),) 

Water level variations in an enclosed basin cannot be estimated satis- 
factorily if a basin is irregularly shaped, or if natural barriers such as 
islands affect the horizontal water motions. However, if the basin is 
simple in shape and long compared to width, then water level elevations 
may be reasonably calculated using the hydrodynamic equations in one 
dimension. Thus if the motion is considered only along the x-axis (major 
axis), and advection of momentum, pressure deficit, astronomical effects 
and precipitation effects are neglected, then Equations 3-50 and 3-52 
reduce to 

3U 98 1 



9s aiu 

at ~ ax 



(3-80) 



If it is further assumed that steady state exists, then Equation 3-79 becomes 

dS 1 



dx pgD 



i^s+'^b) (3-81) 



The bottom stress is taken in the same direction as the wind stress, since 
for equilibrium conditions the flow near the bottom is opposite to flow 
induced by winds in the upper layers. Theoretical development of this 
wind setup equation was given by Hellstrom (1941), Keulegan (1951), and 
others. The mechanics of the various determinations have differed some- 
what, but the resultant equation has been about the same. This wind 
setup equation is expressed as: 

k'np^W'F 

AS = COS0 (3-82) 

PgD 

3-128 



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3-129 



where 

k' = a numerical constant as 0.003 

p^ = air density 

W = Wind velocity 

F = Fetch length 

G = angle between wind and the fetch 

n = 1 + '^s/'^b; 1.15 < n < 1.30 

AS = Wind setup. This value represents 
the difference in water level 
between the two ends of the fetch 

D = average depth of the fetch . 

An approximate expression of Equation 3-82 is given by 

CW^F 

AS = cos 6 (3-83) 

D 

where C is a coefficient having dimensions of time squared per unit 
length. Saville (1952) in a comprehensive investigation of setup data 
obtained from Lake Okeechobee found that C is approximately 1.165 x 10"^ 
when W is given miles per hour, F in miles and D and AS in feet. 
This coefficient is almost identical with that of the Netherlands Zuider 
Zee formula (1.25 x 10"^). 

Equation 3-83 is often useful in making the first approximation of 
the setup in an enclosed basin. Its advantage is that the setup can be 
evaluated with fewer computations. The surge can be estimated more satis- 
factorily by segmenting an enclosed basin into reaches and using a numeri- 
cal integration procedure to solve Equation 3-81 for the various reaches 
in the basin. Bretschneider (Ippen, 1966) presented solutions in parameter 
form for Equation 3-81, and compiled these solutions in tables for different 
conditions. These tables can be used to estimate the storm surge for a 
rectangular channel of constant depth with either an exposed bottom or a 
nonexposed bottom and a basin of regular shape. Such solutions may provide 
an estimate of the water level variation in some basins. 

A more complete scheme than those described above follows. This 
method accounts for the time dependence of the problem; more specifically 
Equations 3-79 and 3-80 are used. Although this more complete method 
provides a better approximation of the surge problem, it is done at the 
expense of increasing the computations. 



3-130 



When a basin enclosed with vertical sides has a well-defined major 
axis and a gradually varying cross section, it is possible to use a 
dynamic one-dimensional, computational model for evaluating fluctuations 
in water level resulting from a forcing mechanism such as wind stress, 
and also to account for some of the effects of a varying cross section. 
The validity of such a model depends on the behavior of the storm system 
and the geometric configuration of the basin. 

Equations 3-79 and 3-80, when the varying width b of the basin is 
introduced, can be written in terms of the volume flow rate Q(x,t) as 

9Q 9S b , 



as _ 1 aq 

9t b dx 



(3-85) 



where x is taken along the major axis of the basin, and for any time t, 
A is the cross-section area, and D is the average depth A/b or 
D = d + S. 

Various schemes have been proposed for evaluating the water level 
changes in an enclosed body of water by using the differential equations. 
(Equations 3-84 and 3-85.) The formulation of the problem and the numer- 
ical scheme given here is from Bodine, Herchenroder and Harris (1972) . 

The surface stress and bottom stress terms are taken identical to the 
terms given for the quasi-static method for open-coast surge. (See 
Equations 3-55 and 3-56, Section 3-865b(l) (a) .) The stress terms, in 
terms of the volume flow rate, become 



(3-86) 



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s 

p 


kww^ = 


kw^ 


cos 


d 



(3-87) 



Substituting Equations 3-86 and 3-87 into Equation 3-84 gives 

— = bkWW, - gA ^ - -^^ (3-88) 



3-131 



A finite difference representation of Equations 3-88 and 3-85 may be 
expressed in the form 



Qi'l = ^ [qi, + f (b,,,/, +b,,3/,) (kw^ cose):';^ 



(3-89) 



At 



where , 






n + 1 



G = 1 + 



{^I'A-'^ls/S (V/.+V3A) 



(3-90) 



(3-91) 



The value of G is greater than unity for most flow conditions except 
for the case when the flow vanishes (Q = 0). The subscripts and super- 
scripts i and n are used to denote discrete points in space and time, 
respectively. A schematic of the grid system used is shown in Figure 3-58. 
It is seen that S at the new time level (t + At) is first evaluated 
based on the known values of Q, D, S, A and b(x) lying on triangle (1) 
and subsequently followed by an evaluation of S at the new time level 
based on known values lying on triangle (2). The solutions at successive 
time levels are obtained by a marching process with Q evaluated at the 
new time level for all integer steps along x; S is evaluated for all 
mid-integer steps along x. Width as a function of x, b(x), is taken 
constant for all t and the total depth D is permitted to vary with 
time. Thus the cross-section area of the basin A is a function of 
distance along the major axis of the basin and a function of time. 

The computational scheme is a combined boundary and initial value 
problem. At each end of the basin, it is assumed that there is no flow 
across the boundary, thus Q = at the boundary. The initial conditions 
assumed are that Q = and S is uniform throughout the basin. 

The scheme requires that for numerical stability, the time increment 
specified for successive calculations At be taken less than Ax/i/g^ax> 
where Dmax is the maximum depth (d + S) anticipated in the basin during 
passage of a storm system. (Abbott, 1966.) 

The restriction imposed by the criterion for numerical stability 
results in a trade-off between the resolution obtained in the solution 
and the number of calculations involved. Decreasing Ax gives better 
resolution of the surge, but requires a smaller At and increases the 
number of computational steps. It is important to choose Ax small 
enough so that reasonable resolution of the surge is obtained, but large 
enough to reduce the computations. The choice of a Ax depends on the 
problem involved. 



3-132 







CM 
0> 



O 

I 



O 



X 

e 

c 

■D 

~ e 

— o 



X 



< 



X 



c 



X 

<a 

I 

I 



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0) 

+J 

M 
X 

CO 

•H 



00 

I 
4) 

E 

■H 



3-133 



For such a recursive type calculation, computer computations reduce 
the effort required. However, since this is a one-dimensional formulation 
of the problem, it is possible to carry out the necessary computations 
manually. 

The March 1955 storm on Lake Erie is used to demonstrate the scheme. 
A plan view of Lake Erie is shown in Figure 3-59. The width and cross- 
section area of the Lake are shown in Figure 3-60, mean bottom profile in 
Figure 3-61, and wind and wind direction data for the March 1955 storm in 
Figure 3-62. 

In the following example problem computations are made at a single 
spatial point where Q is evaluated at a distance of 3Ax from Buffalo 
(Fig. 3-59) at a time when the wind speeds are approximately maximum. 
Tables similar to Table 3-10 should be used in manual calculations. 



*********** EXAMPLE PROBLEM 



GIVEN: 



X 


= 


10 miles 


t 


= 


0.2 hrs. 


g 


= 


79000 mi/hr2 


K 


= 


0.003 



The wind at the new time level is 50.5 miles per hour and 9=4.0. 

The corresponding water surface widths for this section are h-i+i/z = 26.3 



miles and hi+3/2 =21.0 miles, 
the previous time level are 



The values from preceding calculations at 





= 


0.0707 mi?/hr. 


s" 
^+l/2 


= 


3.32 ft. 


^i+3/2 


= 


3.76 ft. 


^-/■l/2 


= 


0.426 mi? 


'^i+z/2 


= 


0.278 mi? 




= 


0.0162 mi. 


^i+l/2 


= 


0.0132 mi. 



Also required is the value of Q. from the previous spatial step which 
is given as +0.0915 mi?/hr. 



3-134 



o 
_l 
< 




u 



I 

to 



o 



3-135 




20 40 60 80 100 120 140 160 180 200 220 240 260 

Distonce from Toledo ( Statute Miles) 

(after Plotzmon ond Roo, 1963) 



Figure 3-60. Cross-Section Area and Surface Width -Lake Erie 



3-136 




20 40 60 80 100 120 140 160 180 200 220 

Distance from Toledo ( Statute Miles) 

Figure 3-61. Mean Bottom Profile of Lake Erie 



240 260 



3-137 





1 ] ' Q . ■ . L^ • ! i , i 1 'h'-^. : 1 !_!_ , J_ i_ -J [ Li- i ■ >_ ^ 




"ui : ! ! i 1' , i 1 : ±: ri- "*■*<,_ ' : 1 ; \ ^^-HT i. 4 


-•< h i "■"" i i -l--l-t- ' ' '•'^ i : > ! /i i 1 : 


■ ! i 1 ■ 1 i ■ ■ ' "^ *M ■ i : ! 'Til' . . .J ^...;.. 


■■'■ ' __^_ ■ ' J I ^_ i 1 i ■■ , , M » : L^ 1 1 


' "TT- i ! : -r' -p '1 1 -T-!- ^^ _. ■ , XT ; 4IU ' : 


• 1 ; ; U I ■ _1_ M ^e-^^^ ._.^; ^-^^ 1 ■ 1 . 1 


■-4— -'-t- -1-4- ' 4---t- -:^' • . ..-^'-i-^-^ -f -4- 


_J_p _U ^u;-r- -- -t-f+^-l^-J^H'^^iS^-J— — 4--^--^ -^ ^-H: 


" i- ' i ^ i i \' M '^.^ ■ II II j 


-+-T- -] — i ; III 1 Vi ■ V , 1 1 ! , ! 




1 i 1 i : 1 i 1 1 Wr"^ ^ ' ! 1 ; i i 1 ! 


1 1 1 1 , ' ! ' 11 i 1 1*: i ■ 1 -rti ' 1 ■ 


-r - " i- ■ - 1 i 1 ! : i M • 1 i\ ' : 1 1 'g ' 1 1 




Xdzt 't 4 Z -J ' -^ -> .- ! , 1 ' ' S ' i ' i i _i- ■ - 


I ' 1 i 11' 1 ^ i 1 1 i ' ' 


1 ^ -^-y ■ ; "i ! - ; !5 1 ; 1 1 I , ...i . 


__^ LJ ^__i-.H_j____ u^^^AJL^ ^.^^_4--L4- 


-H-' ^--l--U_Lt rV i ■ 1 -t— H -"-—-J -^— iTTi — ^— 1 4— 


I'T "I 'l-i U hI I-(-L ' , i 1 J . L 1 ^+- 1 ' a. I .,-.^:JLlL.- 1 _U_f__U:. 


r- ■■ -r"- -^f- -^ 1 -^ ^^*T^ '1 111 :<B 1 : 1 1 i :_ 


J jl ^_l _i_ ' _L Lft^ , 1 _^ 1 ■ 1 i ! *r J ; ,_, 


i -^ zL"""t ! SL _,, _ -._ 1- . . uj L-L- .1- iS-i ' U.L-. 


i i 111 1 ^ , ., i : 1 1 : J r ! 




ibJ-r3_ip..z_±;:4:±|:i4:4:-^,M±L:itiLj==LJ=t--C^4i^ Ljr_ 


-4-4-1- -1-' 4- -1- -H- -rr =^ -H^ H — ^ : ' ' 1 -t-H !-!^'-i- -i — ri- ' ^ 






H- ' ' ' - - ■ p'-i.i 1 i 1 i 1 j > t : ! 1 :o» !5 , ... i...! i ij 


j; ■■■•-■ • ",' _L, ■* ' 1 1 _i_ i -^' ' I I ^ -1 U' ! ' "i .^ J 1 1 . i 


' -j- ■ 1 ' L. "T ' ■ ~^^~^ ■ i J ' M ; 1 ' 1 I* ^ I 1 ! .Ill 


■■ ■ : i -±- : \ I ^^ ' ' I'M 1 ! _La ^ .; ' ■ _^ • i m r^ p i i _,_ , i , l 




I i 1 1 !_;. ■ - -1 i " i 1 z' i ■ ■ ill' ■ 


-v-r*^ 5;-T-l- - -i~r-rf- H — :~- -TT-' 1 : ' ' ^f+-| hT+-^ ■•-'-t-H- ' p 1 Ml, ; 1 : '- ■ 




"-L'- ^ : M ■ I'^LIM^-'— 'Li' ' • ■■ ' '• - 


xtLT ' • r^Srr 1 — m . . . ■ x \ ■• 1 i ' ■ \ < > 1 g i ■■ m 


1 I ,1/. 1 M MM •! ' ■ M M , M : 1 


_ ' . 4_^.-_gjL! _[_uLL ■ ; : ; _^__Jl-u-^ _Mej_ M 1 i . . , ; ■ __^. 


— ^ ^>^g^r^-' ! i ■ ■ M : ^^^--i^-- H--T-~i-- r^— 




Mm ~r^s MM "li-;-' M : , -^^^ 1 1 ,Q- , M MM ■ ' _j_ : , ! L 


Ti . i -^-^ ^ SZMmL '":'"t-r" : 1 r M ! ' 1 J i I ■ W> '1 1 1 . 1 ! : 1 ; 


M : _ C&'T; Li M ! ■ M -t MT-j 1-/ ■ M 1 L: m _i.^. i ; ; . 


q 1 ,~r i^::i_L.-.- >isq..4 mm i m ^.^i. , . 1 — , S . -__^j mi u^ — ^ -mi_m 




'^ ^' M Ml Ml ! i ^ 1 1 ' IT i ' . i ■ i ■ i 1 M 


— 1 — ^-MZi:p=!44,a;;;j-iM--t-r-H- i> M M h -4^+^^^ H[ M 1 1 -j-H-j- 






; 1 1 ' ' i ' ' i ^' 1 1 ' 1 1 ' 1 1 i 1 !/ ' i ' ! 1 i i 1 i 1 


■ ' ' ' 11 " _^^r^ 1 ' ■ ■ 11 1 i ! 1 1 / i 1 1 ' ' '< ■ ■ ' ' i 




M I ^ M -H — t- , ; M , . ,i~' ■ — 'H— +--wn_ f'-'Mj-J — _ ■ ' 1 - 

, ■ 1 1 1 M M -»' . . , i I M < ' 1 1 M i 1 . . ■ 


z — n ^.-*-r "T-i{i 1 M • ,j^ ■ ' i 1 i 1 1 i I . -i-Mj ■ ■ M 




! I! "M i 1 ■ T- l>'i ■ - . 1 rr • M , ■ M : : i / . i 1 'Ml . . 1 M , . 


-1 — TT" .-^-^-r Mraz T* ~ 1 ' i : m , fsj mm .11 ■ m : , , ■ 1 


Mil v^ M ! ; i : I ■ ■ M 1 r<L ' ' ' ' ^ ' 


■ 1 1 —-;■■' ,r : 1 t ^ M ' Ml- MM ■ i '^■^M I M | i i m . . : ; : 


M ! 1 • . 1 1 1 1 M M' i III 111! 11- 111 ' '"^ 1 ! 1 j ■ '■ 1 ■ ■ 


M , 1 "^ r- ifiT-! -yi 1 11 ' M . 1 ' 1 ! i \'7: 1 1 ; mi _i_4_i 


! 1 M i ;| M vr M j MM 1,1 ill 1 X, 1 1 M ' ' ' i ' ' 


MM M -'M ' ^ 1 M ' Ml! i 1 1 Ml 1 X. i M Ml 


_M ' ' ■ 1 I -rM-T- 1 1 1 1 ^ 1 1 M M ! ^^M i ^ M ' ' ^ >L m m m 1 m _ 


~M 'iLm,'''m :" MM Mil , ! 1 ' ' : M -^ II MM 1 


pM— M — t^ 1 rn n r Ml!,: 1 1 1 I 1 7^ : , i m , i i 


' u' 1 1 i 1 r Ml"! ; i M 1 I _i_ _LJ u-<<^-i-i- ' 


-^ _i.^fz l^ zr — !~ 1 M M '■■'•■ ^^ i 1 ^ >^ 1 , i 1 ; _La 


zznif'^ lt±1_i_ _i_ 1 h' 1 "!'•' i"' 1 1 1 M yr 1 ■ i i i 1 1 


>f'| ! 1 1^' 1 M 1 ' M<Li ' • '■ 


|! M ' -^ : i ! ' ' ' 1 T*"*-Li = 


_i_ [L. , 1 1 IM- ■:^ : -^ n_ ' H-(- ' 111! '4\. . 1 ,^-i.^. 


-^ it zm -^ ^"'^ 1" m£4- l m M -U- 4 , ' ' ■-i--A- ..j ..1 i L 


■ \ -~p i 1 n~ ^~m M"Sj L i i ...J L.I , III (>^ ^ [-^ ' 1 


hM ■, Mi i, I • M M ■ M ' i*«-,L' ' 


I 1 1 1 / 1 1 M Ml I M 1 ; ^-isj 11- 1 J 


r Is 1 1 i 1 L^ ' 1 ' M ■ / ' ■ ' n 


'V_U ^ ""U_LJv 'M- LJ-^. i > -A I - 1 ! A 


^g- znt I r_i _ . 1 M .L -L- . . i. 1 i_ >«),J--L- 


jT j>zJ "" ■ _JM7~"i"L; ^ !-!-U i -j - ■ ■ ' ' -LjZ^^it ' 1 


p rT'~'' rtlM M n iT'i M -r-i 1 M 1 1 [Mil >4 1 


rU i/i 1 i 1 M ' 1 ' '■ iV 


1 1 "*■■>•-]• #, MM 1 , 1 \ i : iU,-' iSt^- 


_i- M • T r^i. L M 1 ~^ _i ! 1 L . -4 _i _i. 1 M .' i -J i -u u^-u 


l: j I M T''«fi;.n" 1 "^ 1^1 1 L-L-.. M 1 ; i ; 1 ' 


M iM-i n hri "ri^^-GiM" i m 1 t ^_L_L 1 : . aJ 



o 
o 



o 








o 








00 






in 


o 


■o 




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CM 


to 

CM 




a 

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CO 


o 






1 


u> 








o 






1— 

LU 
a> 
o 


o 






_l 


o 








^a- 




V) 


o 


CM 




^ 
3 








O 


c= 






X 


o 






•—^ 


^_ 






<l> 


o 






E 


0) 


r-) 








o 




f- 


Q 


on 














■o 
c 
o 

■o 
a> 


o 


■D 




Ci. 


o 






CO 


CM 


CM 
CM 




■o 

tz 


o 






to 


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1 


(O 






ro 


o 






Q> 



CM 



o in o in O 

CO in m ^ ^ 



in o lo o 

ro ro CM CM 

(HdW) paadSpujM 



in 



8^ 

O® CM 



























3S!M)|00|DJ8|UnO3 










as!M>)oo|3 






1 

o 

O 

00 


1 

o 

o 
in 


1 1 1 

o o o 
O O O 
CM 0> CD 


1 
o 
O 
ro 


o 
C 


1 
o 
> O 
(O 




1 1 1 

o o o 
O O O 

CO a> CM 


1 

o 

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in 


1 
o 
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I I 

(saajBaQ ) 3V|di ^o sjxv M*!« a|6u\/ 



3-138 



FIND: Wind setup at the new time level. 

SOLUTION : The wind stress coefficiei t given by Equation 3-59 is 

k = K, + K, (l - ^) = 1-1 X 1«" + 2.5 X 10- (l-^^j 
= 2.27 X 10"* 

From Equation 3-91 

4KAt|Q;'^^ I 



G = 1 + 



(D?.:^ + D?+3/J"{b,.^j,+b,>3/J 



4 (0.003) (0.2) I 0.0707 I = i oi 

*^ ~ "^ "^ [(0.0162)2 + (0.0132)2] (26.3 + 21.0) 

The two terms needed in the evaluation of Q, (Equation 3-89) are 

given by 



f (b.>'^+b,,3/^) (kW^ COS 0):';/ 



0-2 

(26.3 + 21.0) (2.27 XIO'^) (50.5)^ (1) = 0.0274 



and 



gAt 



{^i+'A '^ ^i+3/2) {^i+'A ^1 + 3/2)" 



= - 0.0463 



2Ax 

79,000 (0.2) (0.426 + 0.278) (3.32-3.76) 
(2) (10) (5,280) 

The volume flow rate from Equation 3-89 is 



QLV = ~ [0.070 + 0.0274-0.0463] = 0.0513 mi? /hr. 

^^ 1.0129 



3-139 



and finally, the water level at this discrete point in space and time 
is given by Equation 3-90 as 



At / _ _ \n+l 
i+'A 
(0.2) (0.0915 -0.0513) (5280) 



^"4 = C«-^(Q,-Q«)' 



= 3.32 + 

(10) (26.3) 

= 3.48 feet, say 3.5 feet . 

The significant digits indicated in the above computations do not 
reflect the accuracy of the numerical procedure, but are retained to 
reduce the accumulation of round-off errors. 

The wind setup hydrograph for the ends of the lake as determined by 
computer is shown in Figure 3-63. The storm winds blew in the general 
direction toward Buffalo at the northeastern end of the lake as indi- 
cated by the setdown at Toledo and the setup at Buffalo. The spatial 
steps Ax taken are quite large; smaller increments in Ax would give 
a more accurate estimate. Calculations with the mathematical model 
were initiated with the system in a calm state, i.e., Q = S = 0. 

The wind setup profile along the major axis of the lake determined 
by the numerical scheme is shown in Figure 3-64 for three time periods 
during the storm. The nodal point where the computed water surface 
crosses the Stillwater level occurs near the center of the lake, but 
this nodal point can vary with time. 

Although the comparison of the computed and observed wind setup is 
not in complete agreement, particularly at the beginning and late stages 
of the storm, the method gives reasonable results for the wind setup 
amplitudes. To engineers, it is frequently the maximum departure of 
the water level from its normal position that is of greatest concern. 
Results of the simplified model should be interpreted with care, since 
many of the physical processes which may be significant have been neg- 
lected. Wind and bottom stress laws, in particular, are oversimplified 
for the Lake Erie problem. Better agreement can be expected with two- 
dimensional schemes such as the one developed by Platzman (1963) , since 
they more accurately model for the physical processes involved. 



(3) Storm Surge in Semienclosed Basins . It is generally im- 
possible to make reliable estimates of storm surge in semienclosed basins 
(bays, and estuaries) with less exact procedures such as those described 
for specific problems within enclosed basins or on the open coast. This 
is because bays and estuaries are nearly always irregular in shape, and 
basin geometry is often further complicated by the presence of islands, 
navigational channels, and harbors. Moreover, many of these basins have 



3-140 




in 



o 

t— 

o 

e 

I— 
O 

-^- 

en 

I 
o 

T3 
(1> 



T3 

C 

o 

o 
o 

M— 

CD 



O 

t— 

o 
■o 

X 



CO 
T3 

c 



ro 

CO 

ro 



(^aaj) dnias pujM 



3-141 



oo 




1/3 






o 
•p 

CO 



u 

m 

0) 

o 



•H 

o 
a> 

CO 

c 



I 

to 

O 
U 

3 
bo 

•H 
[I. 



(O 



<\j O cvj ^ 

I I 

(499j)dnias puiM 



I 



CO 
I 



o 
I 



3-142 



extensive low- lying areas surrounding them which may be subject to exten- 
sive flooding during severe storm conditions. Also, these basins are 
usually shallow, and on the windward side of the basin substantial bottom 
areas can become exposed. Thus, during the passage of a storm system, 
many semienclosed basins have time-dependent moving boundaries. The 
water present at any time in a bay or estuary is also dependent upon the 
sea state because of the interrelation between these two bodies of water. 
Because of the above characteristics of semienclosed basins, a simplified 
approach does not usually provide a satisfactory estimate of water motions, 
and methods based on full two-dimensional dynamic approaches should be 
employed. 



3-143 



REFERENCES AND SELECTED BIBLIOGRAPHY 

ABBOTT, M.B., An Introduction to the Method of Charaateristias^ American 
Elsevier, New York, 1966, 243 pp. 

ARAKAWA, H., and SUDA, K. , "Analysis of Winds, Wind Waves, and Swell Over 
the Sea to the East of Japan During the Typhoon of September 26, 1935," 
Monthly Weather Review ^ Vol. 81, No, 2, Feb. 1953, pp. 31-37. 

BAER, J., "An Experiment in Numerical Forecasting of Deep Water Ocean 
Waves," Lockheed Missile and Space Co., Sunnyvale, Calif., 1962. 

BARNETT, T.P., "On the Generation, Dissipation, and Prediction of Ocean 
Wind Waves," Journal of Geophysical Research^ Vol. 73, No. 2, 1968, 
pp. 531-534. 

BATES, C.C, "Utilizations of Wave Forecasting in the Invasions of 
Normandy, Burma, and Japan," Annals of the New York Aaademy of 
Sciences J Vol. 51, Art. 3, "Ocean Surface Waves," 1949, pp. 545-5 72. 

BELLAIRE, F.R., "The Modification of Warm Air Moving Over Cold Water," 

Proceedings 3 Eight Conference on Great Lakes Research, 1965. 

BODINE, B.R., "Storm Surge on the Open Coast: Fundamentals and Simpli- 
fied Prediction," TM-35, U.S. Army, Corps of Engineers, Coastal 
Engineering Research Center, Washington, D.C., May 1971. 

BODINE, B.R., HERCHENRODER, B.E., and HARRIS, D.L., "A Numerical Model 
for Calculating Storm Surge in a Long Narrow Basin with a Variable 
Cross Section," Report in preparation, U.S. Army, Corps of Engineers, 
Coastal Engineering Research Center, Washington, D.C. , 1972. 

BORGMAN, L.E., "The Extremal Rayleigh Distribution: A Simple Approxima- 
tion for Maximum Ocean Wave Probabilities in a Hurricane with Varying 
Intensity," Stat. Lab Report No. 2004, University of Wyoming, Laramie, 
Wyo. , Jan. 1972. 

BOWEN, A.J., INMAN, D.L., and SIMMONS, V.P., "Wave 'Setdown' and Set-up," 

Journal of Geophysical Research, Vol. 73, No. 8, 1968. 

BRETSCHNEIDER, C.L., "Revised Wave Forecasting Curves and Procedures," 
Unpublished Manuscript, Institute of Engineering Research, University 
of California, Berkeley, Calif, , 1951. 

BRETSCHNEIDER, C.L. "Revised Wave Forecasting Relationships," Proceedings 
of the Second Conference on Coastal Engineering, ASCE, Council on 
Wave Research, 1952a. 

BRETSCHNEIDER, C.L., "The Generation and Decay of Wind Waves in Deep 
Water," Transactions of the American Geophysical Union, Vol. 33, 
1952b, pp. 381-389. 



3-145 



BRETSCHNEIDER, C.L., "Hurricane Design-Wave Practice," Journal of the 
Waterways and Harbors Division^ ASCE, Vol. 83, WW2 , No. 1238, 1957. 

BRETSCHNEIDER, C.L., "Revisions in Wave Forecasting; Deep and Shallow 
Water," Proceedings of the Sixth Conferenoe on Coastal Engineering, 
ASCE, Council on Wave Research, 1958. 

BRETSCHNEIDER, C.L., "Wave Variability and Wave Spectra for Wind- 
Generated Gravity Waves," TM-118, U.S. Army, Corps of Engineers, 
Beach Erosion Board, Washington, D.C., Aug. 1959. 

BRETSCHNEIDER, C.L., "A One Dimensional Gravity Wave Spectriim," Ooean 
Wave Spectra, Prentice-Hall, New York, 1963. 

BRETSCHNEIDER, C.L., "Investigation of the Statistics of Wave Heights," 
Journal of the Waterways and Harbors Division, ASCE, Vol. 90, No. WWl, 
1964, pp. 153-166. 

BRETSCHNEIDER, C.L., "Generation of Waves by Wind; State of the Art," 
Report SN-134-6, National Engineering Science Co., Washington, D.C., 
1965. 

BRETSCHNEIDER, C.L., Department of Ocean Engineering, University of 
Hawaii, Honolulu, Hawaii, Consultations, 1970-71. 

BRETSCHNEIDER, C.L., "A Non- Dimensional Stationary Hurricane Wave Model," 
Offshore Technology Conference, Preprint No. OTC 1517, Houston, Tex., 
May, 1972. 

BRETSCHNEIDER, C.L., and COLLINS, J. I., "Prediction of Hurricane Surge; An 
Investigation for Corpus Christi, Texas and Vicinity," Technical Report 
No. SN-120, National Engineering Science Co., Washington, D.C., 1963. 
(for U.S. Army Engineering District, Galveston, Tex.) 

BRETSCHNEIDER, C.L., and REID, R.O., "Change in Wave Height Due to Bottom 
Friction, Percolation and Refraction," S4th Annual Meeting of American 
Geophysical Union, 1953. 

BUNTING, D.C., "Evaluating Forecasts of Ocean-Wave Spectra," Journal of 
Geophysical Research, Vol. 75, No. 21, 1970, pp. 4131-4143. 

BUNTING, D.C., and MOSKOWITZ, L.I., "An Evaluation of a Computerized 
Numerical Wave Prediction Model for the North Atlantic Ocean," 
Technical Report No. TR 209, U.S. Naval Oceanographic Office, 
Washington, D.C., 1970. 

CHRYSTAL, G., "Some Results in the Mathematical Theory of Seiches," Pro- 
ceedings of the Royal Society of Edinburgh, Session 1903-04, Robert 
Grant and Son, London, Vol. XXV, Part 4, 1904, pp. 328-337. 



3-146 



CHRYSTAL, G., "On the Hydrodynamical Theory of Seiches, with a Biblio- 
graphical Sketch," Transactions of the Royal Society of Edinburgh, 
Robert Grant and Son, London, Vol. XLI, Part 3, No. 25, 1905. 

COLE, A.L., "Wave Hindcasts vs Recorded Waves," Supplement No. 1 to Final 
Report 06768, Department of Meteorology and Oceanography, University 
of Michigan, Ann Arbor, Mich., 1967. 

COLLINS, J.I., and VIEW4AN, M.J., "A Simplified Empirical Model for Hurri- 
cane Wind Fields," Offshore Technology Conference, Vol. 1, Preprints, 
Houston, Tex., Apr. 1971. 

CONNER, W.C., KRAFT, R.H., and HARRIS, D.L., "Empirical Methods of Fore- 
casting the Maximum Storm Tide Due to Hurricanes and Other Tropical 
Storms," Monthly Weather Review, Vol. 85, No. 4, Apr. 1957, pp. 113-116. 

COX, D.C., ed. , Proceedings of the Tsunami Meetings Associated with the 
10th Paoifia Science Congress, Honolulu, Hawaii, 1961. 

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3-157 



CHAPTER 4 



LITTORAL 



PROCESSES 





'^ 



\ 






,^^ 



.' './•»- 



NORTH CAROLINA COAST - 13 February 1973 



CHAPTER 4 

LITTORAL PROCESSES 

4.1 INTRODUCTION 

Littoral processes result from the interactions among winds, waves, 
currents, tides, sediments, and other phenomena in the littoral zone. The 
purpose of this chapter is to discuss those littoral processes which involve 
sediment motion. Shores erode, accrete, or remain stable, depending on the 
rates at which sediment is supplied to and removed from the shore. Excessive 
erosion or accretion may endanger the structural integrity or functional 
usefulness of a beach, or other coastal structures. Therefore an under- 
standing of littoral processes is needed to predict erosion or accretion 
rates. An aim of coastal engineering design is to maintain a stable shore- 
line where sediment supplied to the shore balances that which is removed. 
This chapter presents information needed for understanding the effects of 
littoral processes on coastal engineering design. 

4.11 DEFINITIONS 

In describing littoral processes, it is necessary to use clearly 
defined terms. Commonly used terms, such as "beach" and "shore," have 
specific meanings in the study of littoral processes, as shown in the 
Glossary. (See Appendix A.) 

4.111 Beach Profile . Profiles perpendicular to the shoreline have char- 
acteristic features that reflect the action of littoral processes. (See 
Figure 1-1, Chapter 1, and Figures A-1 and A-2 of the Glossary for spe- 
cific examples.) At any given time, a profile may exhibit only a few 
specific features, but usually a dune, berm and beach face can be identi- 
fied. 

Profiles across a beach adapt to imposed wave conditions in ways as 
illustrated in Figure 4-1 by a series of profiles taken between February 
1963 and November 1964 at Westhampton Beach, N.Y. The figure shows how 
the berm built up gradually from February through August 1963, then was 
cut back in November through January, and was rebuilt in March through 
September 1964. This process is typical of a cyclical process of storm 
erosion in winter and progradation with the lower, and often longer, 
waves in summer. 

4.112 Areal View . Figure 4-2 shows three generalized charts of different 
U.S. coastal areas, all to the same scale. Figure 4-2a shows a rocky coast, 
well-indented, where sand is restricted to local pocket beaches. Figure 
4-2b shows a long straight coast with an uninterrupted sand beach. Figure 
4-2c shows short barrier islands interrupted by inlets. These are some of 
the different coastal configurations which reflect differences in littoral 
processes and the local geology. 

4.12 ENVIRONMENTAL FACTORS 

4.121 Waves . Water waves are the principal cause of most shoreline 
changes. Without wave action on a coast, most of the coastal engineering 



4-1 



Datum = MTL = Mean Tide Level 
Initial Survey 11 October 1962 

- MTL-Beach Intersect 
(Surveys did not always reach MTL) 



20 



a> 
a> 



o 10 




-300-200 -100 100 200 300 
Distance from Mean Water Line of Initial Survey, feet 

Figure 4-1. Typical Profile Changes with Time, Westhampton 
Beach, New York 



4-2 




Rhode Island 
(See USCaCS Chart I2IQ). 




OCEAN 



a. Rocky Coast with Limited Beaches 



Northeastern Florida 
(See use 8 GS Chart 1246) 



BAY 




OCEAN 

b. Straight Barrier Island Shoreline 



Southern New Jersey 
(See use acs Chart 1217) 



BAY 




c, Short Barrier Island Shoreline 



Figure 4-2. Tiiree Types of Shoreline 



4-3 



problems involving littoral processes would not occur. A knowledge of 
incident waves, or of surf, is essential for coastal engineering planning, 
design, and construction. 

Three important aspects of a study of waves on beaches are: the 
theoretical description of wave motion; the climatological data for waves 
as they occur on a given segment of coast; and the description of how 
waves interact with the shore to move sand. 

The theoretical approach can provide a useful description of water 
motion caused by waves when the limiting assumptions of the theory are 
satisfied. Surprisingly, the small -amplitude theory (Section 2.23) and 
aspects of solitary wave theory (Section 2.27) have proved useful beyond 
the limits assumed in their derivations. 

The theoretical description of water-wave motion provides estimates 
of water motion, longshore force, and energy flux due to waves. These 
estimates are useful in understanding the effect of waves on sediment 
transport, but currently (1973) the prediction of wave-induced sediment 
motion for engineering purposes relies heavily on empirical coefficients 
and judgement rather than on theory. 

Statistical distributions of wave characteristics along a given shore- 
line provide a basis for describing the wave climate of a coastal segment. 
Important wave characteristics affecting sediment transport near the beach 
are height, period, and direction of breaking waves. Breaker height is 
significant in determining the quantity of sand in motion; breaker direc- 
tion is a major factor in determining longshore transport direction and 
rate. Waves affect sediment motion in the littoral zone in two ways: 
they initiate sediment movement, and they drive current systems that trans- 
port the sediment once motion is initiated. 

4.122 Currents . Water waves induce an orbital motion in the fluid beneath 
them. (See Section 2.23.) These are not closed orbits, and the fluid 
experiences a slight wave-induced drift, or mass transport. Magnitude and 
direction of mass transport are functions of elevation above bottom and 
wave parameters (Equation 2-55), and are also influenced by wind and tem- 
perature gradients. The action of mass transport, extended over a long 
period, can be important in carrying sediment onshore or offshore, par- 
ticularly seaward of the breaker position. 

As waves approach breaking, wave-induced bottom motion in the water 
becomes more intense, and its effect on sediment becomes more pronounced. 
Breaking waves create intense local currents (turbulence) that move sedi- 
ment. As waves cross the surf zone after breaking, the accompanying fluid 
motion is mostly uniform horizontal motion, except during the brief pass- 
age of the breaker front where significant turbulence occurs. Since wave 
crests at breaking are usually at a slight angle to the shoreline, there 
is usually a longshore component of momentum in the fluid composing the 
breaking waves. This longshore component of momentum entering the surf 

4-4 



zone is the principal cause of longshore currents - currents that flow 
parallel to the shoreline within the surf zone. These longshore currents 
are largely responsible for the longshore sediment transport. 

There is some mean exchange between the water flowing in the surf 
zone and the water seaward of the breaker zone. The most easily seen of 
these exchange mechanisms are the rip currents (Shepard and Inman, 1950), 
which are concentrated jets of water flowing seaward through the breaker 
zone. 

4.123 Tides and Surges . In addition to wave-induced currents, there are 
other currents affecting the shore that are caused by tides and storm 
surges. Tide-induced currents can be impressed upon the prevailing wave- 
induced circulations, especially near entrances to bays and lagoons and 
in regions of large tidal range. (Notices to Mariners and the Coastal 
Pilot often carry this information.) Tidal currents are particularly 
important in transporting sand in shoals and sand waves around entrances 
to bays and estuaries. 

Currents induced by storm surges (Murray, 1970) are less well known 
because of the difficulty in measuring them, but their effects are un- 
doubtedly significant. 

The change in water level by tides and surges is a significant factor 
in sediment transport, since, with a higher water level, waves can then 
attack a greater range of elevations on the beach profile. (See Figure 
1-7.) The appropriate theory for predicting storm surge levels is dis- 
cussed in Section 3.8. 

4.124 Winds . Winds act directly on beaches by blowing sand off the 
beaches (deflation) and by depositing sand in dunes. (Savage and Wood- 
house, 1968.) Deflation usually removes the finer material, leaving 
behind coarser sediment and shell fragments. Sand blown seaward from 
the beach usually falls in the surf zone; thus it is not lost, but is 
introduced into the littoral transport system. Sand blown landward from 
the beach may form dunes, add to existing dunes, or be deposited in 
lagoons behind barrier islands. 

For dunes to form, a significant quantity of sand must be available 
for transport by wind, as must features that act to trap the moving sand. 
Topographic irregularities, the dunes themselves, and vegetation are the 
principal features that trap sand. 

The most important dunes in littoral processes are foredunes - the 
line of dunes immediately landward of the beach. They usually form be- 
cause beachgrasses growing just landward of the beach will trap sand blown 
landward off the beach. Foredunes act as a barrier to prevent waves and 
high water from moving inland, and provide a reservoir of sand to replen- 
ish the nearshore regime during severe shore erosion. 

The effect of winds in producing currents on the water surface is 
well documented, both in the laboratory and in the field. (Bretschneider, 



4-5 



1967; Keulegan, 1951; and van Dom, 1953.) These surface currents drift 
in the direction of the wind at a speed equal to 2 to 3 percent of the 
wind speed. In hurricanes, winds generate surface currents of 2 to 8 
feet per second. Such wind-induced surface currents toward the shore 
cause significant bottom return flows which may transport sediment sea- 
ward; similarly, strong offshore winds can result in an offshore surface 
current, and an onshore bottom current which can aid in transporting 
sediment landward. 

4.125 Geologic Factors . The geology of a coastal region affects the 
supply of sediment on the beaches and the total coastal morphology. Thus, 
geology determines the initial conditions for littoral processes, but geo- 
logic factors are not usually active processes affecting coastal engineer- 
ing. 

One exception is the rate of change of sea level with respect to 
land which may be great enough to influence design, and should be exam- 
ined if project life is 50 years or more. On U.S. coasts, typical rates 
of sea level rise average about 1 to 2 millimeters per year, but changes 
range from -13 to +9 millimeters per year. (Hicks, 1972.) (Plus means 
a rise in sea level with respect to local land level.) 

4.126 Other Factors . Other principal factors affecting littoral processes 
are the works of man and activities of organisms native to the particular 
littoral zone. In engineering design, the effects on littoral processes 

of construction activities, the resulting structures, and structure main- 
tenance must be considered. This consideration is particularly necessary 
for a project that may alter the sand budget of the area, such as jetty 
or groin construction. In addition biological activity may be important 
in producing carbonate sands, in reef development, or (through vegetation) 
in trapping sand on dunes. 

4.13 CHANGES IN THE LITTORAL ZONE 

Because most of the wave energy is dissipated in the littoral zone, 
this zone is where beach changes are most rapid. These changes may be 
short-term due to seasonal changes in wave conditions and to occurrence 
of intermittent storms separated by intervals of low waves, or long-term 
due to an overall imbalance between the added and eroded sand. Short-term 
changes are apparent in the temporary redistribution of sand across the 
profile (Fig. 4-1); long-term changes are apparent in the more rtearly per- 
manent shift of the shoreline. (See Figures 4-3, 4-4, and 4-5.) 

Maximum seasonal or storm- induced changes across a profile, such as 
those shown in Figure 4-1, are typically on the order of a few feet verti- 
cally and from 10 to 100 feet horizontally. (See Table 4-1.) Only during 
extreme storms, or where the available sand supply is restricted, do un- 
usual changes occur over a short period. 

Typical seasonal changes on southern California beaches are shown in 
Table 4-1. (Shepard, 1950.) These data show greater changes on the beach 



4-6 




m 

u 

1-5 

s 

ID 
2 



e 
o 
■t-> 
+-> 
o 

•H 

(U 
2 

C 

o 

•H 

in 
o 

<L> 

C 



o 



to 
I 

•H 



4-7 




m 
U 

(U 

s 
z 



c 

0) 

> 

o 

ca 



o 

■ H 

tn 
o 
U 



C 
o 

•H 
■P 
<U 
^^ 
O 
O 

4) 

c 



U 
O 



I 

u 



4-8 




u 
o 



IX 

•H 
i-l 
0> 

o 

CO 



-s 



w 



I 

3 

•H 



4-9 



Table 4-1 . Seasonal 


Profile Changes on Southern ( 


California Beaches 


T 1 * 


Erosion 


Date 


Locality 


Vertical* 


Horizontal 




(ft.) 


at MWL (ft.) 




Marine Street 


5.9 


24 


27 Nov 45- 5 Apr 46 


Beacon Inn 


8.2 


117 


2 Nov 45-19 Mar 46 


South Oceanside 


5.0 


41 


2 Nov 45-26 Apr 46 


San Onofre 








Surf Beach 


4.2 


43 


6Nov45-12 Apr 46 


Fence Beach 


10.8 


87 


6 Nov 45-12 Apr 46 


Del Mar 


3.4 




2 Nov 45-10 May 46 


Santa Monica Mountains 


6.2 





28 Aug 45-13 Mar 46 


Point Mugu 


6.0 




28 Aug 45-13 Mar 46 


Rincon Beach 


2.7 




22 Aug 45-13 Mar 46 


Goleta Beach 


0.5 




22 Aug 45-13 Mar 46 


Point Sur 






27 Aug 45-14 Mar 46 


South Side 


1.8 






North Side 


6.3 






Carmel Beach (South) 


6.7 





26 Aug 45-14 Mar 46 


Point Reyes 


5.1 




26 Aug 45-16 Mar 46 


Scripps Beach 








Range A 


1.6 


88 


19 Nov 45-29 Apr 46 


Range B 


3.2 


240 


7 Nov 45-10 Apr 46 


Range C 


5.6 


260 


7 Nov 45-10 Apr 46 


Range D 


5.3 


250 


7 Nov 45- 9 May 46 


Range E 


3.3 





7 Nov 45-24 Apr 46 


Range F 


1.5 


67 


7 Nov 45-10 May 46 


Range G 


0.7 


15 


19 Oct 45-10 Apr 46 


Range H 


1.6 


44 


7 Nov 45-24 Apr 46 


Scripps Pier 


3.8 


33 


13 Oct 37-26 Mar 38 




3.6t 


30 1 


26 Mar 38-30 Aug 38 




4.3 


38 


30 Aug 38-13 Feb 39 




3.2t 


32t 


13 Feb 39-22 Sep 39 




2.5 


28 


22 Sep 39-24 Jan 40 




3.41 


34 1 


24 Jan 40-18 Sep 40 




1.2 


12 


18 Sep 40-16 Apr 41 




1.2t 


lit 


16 Apr 41-17 Sep 41 




2.9 


301 


17 Sep 41-29 Apr 42 




1.9t 


16t 


29 Apr 42-30 Sep 42 



(from Shepard 1950) 

* Vertical erosion measured at berm for all localities except Scripps Beach and 
Scripps Pier where the mean water line (MWL) was used. 

t Accretion values. 



4-10 



than are typical of Atlantic coast beaches. (Urban and Galvin, 1969; 
Zeigler and Tuttle, 1961.) Available data indicate that the greatest 
changes on the profile are in the position of the beach face and of the 
longshore bar - two relatively mobile elements of the profile. Beaches 
change in plan view as well. Figure 4-6 shows the change in shoreline 
position at seven east coast localities as a function of time between 
autumn 1962 and spring 1967. 

Comparison of beach profiles before and after storms suggests ero- 
sion of the beach above MSL from 10,000 to 50,000 cubic yards per mile 
of shoreline during storms expected to recur about once a year. (DeWall, 
et al., 1971; and Shuyskiy, 1970.) While impressive in aggregate, such 
sediment transport is minor compared to longshore transport of sediment. 
Longshore transport rates may be greater than 1 million cubic yards per 
year. 

The long-term changes shown in Figures 4-3, 4-4, and 4-5 illustrate 
shorelines of erosion, accretion, and stability. Long-term erosion or 
accretion rates are rarely more than a few feet per year in horizontal 
motion of the shoreline, except in localities particularly exposed to 
erosion, such as near inlets or capes. Figure 4-5 indicates that shore- 
lines can be stable for a long time. It should be noted that the erod- 
ing, accreting, and stable beaches shown in Figures 4-3, 4-4, and 4-5 
are on the same barrier island within a few miles of each other. 

Net longshore transport rates along ocean beaches range from near 
zero to 1 million cubic yards per year, but are typically 100,000 to 
500,000 cubic yards per year. Such quantities, if removed from a 10- to 
20-mile stretch of beach year after year, would result in severe erosion 
problems. The fact that many beaches have high rates of longshore trans- 
port without unusually severe erosion suggests that an equilibrium condi- 
tion exists on these beaches, in which the material eroded is balanced by 
the material supplied; or in which seasonal reversals of littoral trans- 
port replace material previously removed. 

4.2 LITTORAL MATERIALS 

Littoral materials are the solid materials (mainly sedimentary) in 
the littoral zone on which the waves and currents act. 

4.21 CLASSIFICATION 

The characteristics of the littoral materials are a primary input 
to any coastal engineering design. Median grain size is the most fre- 
quently used design characteristic. 

4.211 Size and Size Parameters . Littoral materials are classified by 
grain size into clay, silt, sand, gravel, cobble, and boulder. Several 
size classifications exist, of which two, the Unified Soil Classification, 
(based on the Casagrande Classification) and the Wentworth Classification, 
are most commonly used in coastal engineering. (See Figure 4-7.) The 






o 

•D 
C 
O 



I -r 



o 
o 
(O 



c 

o 

J= 
CO 

> 

_l 

o 
a> 
W 

c 
o 
a> 



c 
o 



o 




Figure 4-6. Fluctuations in Location of Mean Sealevel 
Shoreline on Seven East Coast Beaches 



4-12 



Unified Soil Classification is the principal classification used by engi- 
neers. The Wentworth classification is the basis of a classification 
widely used by geologists, but is becoming more widely used by engineers 
designing beach fills. 

For most shore protection design problems, typical littoral mate- 
rials are sands with sizes between 0.1 and 1.0 millimeters or, in phi 
units, between 3.3 and phi. According to the Wentworth classification, 
sand size is in the range between 0.0625 cind 2.0 millimeters (4 and -1 
phi); according to the Unified Soil Classification, it is between 0.074 
and 4.76 millimeters (3.75 and -2.25 phi). Within these sand-size ranges, 
engineers commonly distinguish size classes by median grain size measured 
in millimeters, based on sieve analyses. 

Samples of typical beach sediment usually have a few relatively 
large particles covering a wide range of diameters, and many small par- 
ticles within a small range of diameters. Thus, to distinguish one 
sample from another, it is necessary to consider the small differences 
(in absolute magnitude) among the finer sizes more than the same differ- 
ences among the larger sizes. For this reason, all sediment size classi- 
fications exaggerate absolute differences in the finer sizes compared to 
absolute differences in the coarser sizes. 

As shown in Figure 4-7, limits of the size classes differ. The 
Unified Soil Classification boundaries correspond to U.S. Standard Sieve 
sizes. The Wentworth classification varies as powers of 2 millimeters; 
that is, the size classes have limits, in millimeters, determined by the 
relation 2 , where n is any positive or negative whole number, including 
zero. For example, the limits on sand size in the Wentworth scale are 
0.0625 millimeters and 2 millimeters, which correspond to 2 "* and 2 ^ 
millimeters. 

This property of having class limits defined in terms of whole number 
powers of 2 millimeters led Krumbein (1936) to propose a phi unit scale 
based on the definition: 

Phi units (0) = — log (diameter in mm.) (4-1) 

Phi unit scale is indicated by writing ^ or phi after the numerical 
value. The phi unit scale is shown in Figure 4-7. Advantages of phi 
units are: 

(a) Limits of Wentworth size classes are whole numbers in phi 
units. These phi limits are the negative value of the exponent, n, in 
the relation 2 . For example, the sand size class ranges from +4 to -1, 
in phi units. 

(b) Sand size distributions typically are near lognormal, so that 
a unit based on the logarithm of the size better emphasizes the small 
significant differences between the finer particles in the distribution. 

4-13 



Wentworth Scale 
(Size Description) 


Phi Units 

0« 


Grain 

Diameter 

d (mm) 


U.S. Standard 

Sieve Size 


Unified Soil 

Classification 

(USC) 


Boulder 


-8 
-6 

-2 

-1 



1 

2 
3 

4 

8 

12 


256 
76.2 
64.0 
19.0 

4.76 

4.0 

2.0 

1.0 

0.5 

0.42 
0.25 

0.125 
0.074 

0.0625 

0.00391 

0.00024 


3 in. 

% in. 

No. 4 

No. 10 

No. 40 
No. 200 


Cobble 


Cobble 


Coarse 


Gravel 


Pebble 


Fine 


Coarse 


Sand 


Granule 


Sand 


Very Coarse 


Medium 


Coarse 


Medium 


Fine 


Fine 


Very Fine 


Silt or Clay 


Silt 


Clay 


Colloid 







» /A = — 1 



= 



log d (mm) 
Figure 4-7. 



Grain Size Scales (Soil Classification) 



4-14 



(c) The normal distribution is described by its mean and standard 
deviation. Since the distribution of sand size is approximately lognormal, 
then individual sand size distributions can be more easily described by 
units based on the logarithm of the diameter rather than the absolute diam- 
eter. Comparison with the theoretical lognormal distribution is also a 
convenient way of characterizing and comparing the size distribution of 
different samples. 

Of these three advantages, only (a) is unique to the phi units. The 
other two, (b) and (c) , would be valid for any unit based on the logarithm 
of size. 

Disadvantages of phi units are: 

(a) Phi units increase as absolute size in millimeters decreases. 

(b) Physical appreciation of the size involved is easier when the 
units are millimeters rather than phi units. 

(c) The median diameter can be easily obtained without phi units. 

(d) Phi units are dimensionless, and are not usable in physically 
related quantities where grain size must have units of length such as 
grain-size, Reynolds number, or relative roughness. 

Size distributions of samples of littoral materials vary widely. 
Qualitatively, the size distribution of a sample may be characterized by 
a diameter that is in some way typical of the sample, and by the way that 
the sizes coarser and finer than the typical size are distributed. (Note 
that size distributions are generally based on weight, rather than number 
of particles.) 

A size distribution is described qualitatively as well-sorted if all 
particles have sizes that are close to the typical size. If all the par- 
ticles have exactly the same size, then the sample is perfeatly sorted. 
If the particle sizes are distributed evenly over a wide range of sizes, 
then the sample is said to be well-graded. A well-graded sample is poorly 
sorted; a well-sorted sample is poorly graded. 

The median (Mj) and the mean (M) define typical sizes of a sample of 
littoral materials. The median size, N^ in millimeters, is the most com- 
mon measure of sand size in engineering reports. It may be defined as 

M^ = d,„ (4-2) 

where dso is the size in millimeters that divides the sample so that half 
the sample, by weight, has particles coarser than the dso size. An equiv- 
alent definition holds for the median of the phi size distribution, using 
the symbol M^,), instead of M^f. 

Several formulas have been proposed to compute an approximate mean 
(M) from the cumulative size distribution of the sample. (Otto, 1939; 



4-15 



Inraan, 1952; Folk and Ward, 1957, McCammon, 1962.) These formulas are 
averages of 2, 3, 5, or more symmetrically selected percentiles of the 
phi frequency distribution, such as the formula of Folk and Ward. 

M0 = ^ (4-3) 

where i|) is the particle size in phi units from the distribution curve 
at the percentiles equivalent to the subscripts 16, 50 and 84 (Fig. 4-8); 
4>^ is the size in phi units that is exceeded by x percent (by dry weight) 
of the total sample. These definitions of percentile (after Griffiths, 1967, 
p. 105) are known as graphic measures. A more complex method - the method 
of moments - can yield more precise results when properly used. 

To a good approximation, the median, Mj is interchangeable with the 
mean, (M) , for most beach sediment. Since the median is easier to deter- 
mine it is widely used in engineering studies. For example, in one CERC 
study of 465 sand samples from three New Jersey beaches, the mean computed 
by the method of moments averaged only 0.01 millimeter smaller than the 
median for sands whose average median was 0.30 millimeter (1.74 phi). 
(Ramsey and Galvin, 1971.) 

The median and the mean describe the approximate center of the sedi- 
ment size distribution. In the past, most coastal engineering projects 
have used only this size information. However, for more detailed design 
calculations of fill quantities required for beach restoration projects 
(Sections 5.3 and 6.3), it is necessary to know more about the size dis- 
tribution. 

Since the actual size distributions are such that the log of the size 
is approximately normally distributed, the approximate distribution can be 
described (in phi units) by the two parameters that describe a normal dis- 
tribution - the mean and the standard deviation. In addition to these 
two parameters (mean and standard deviation), skewness and kurtosis 
describe how far the actual size distribution of the sample departs from 
this theoretical lognormal distribution. 

Standard deviation is a measure of the degree to which the sample 
spreads out around the mean and may be approximated using Inman's (1952) 
definition, by 



'4> 



'84 ^16 

oa. = : , (4-4) 



where i^iqi^ is the sediment size, in phi units, that is finer than 84 per- 
cent by weight, of the sample. If the sediment size in the sample actually 
has a lognormal distribution, then a± is the standard deviation of the 



4-16 



99.9 
99.8 



99.5 



o 
o 
o 



c 

0) 

o 

k_ 

« 
Q. 

> 



E 
o 




1.5 2.0 

Diameter (phi) 



4.0 



I I I 



J I I I I 



1.5 1.0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.15 

Diameter (mm) 



0.1 0.08 0.06 



Figure 4-8. Example Size Distribution 



4-17 



sediment in phi units. For perfectly sorted sediment, 0^ = 0. For 
typical well-sorted sediments, o, =» 0.5. 

The degree by which the phi size distribution departs from symmetry 
is measured by the skewness (Inman, 1952), as 

^<t> " ^d4> (4-5) 

* a<l> 

where M^ is the mean, I^cf) is the median, and a^ is the standard 
deviation in phi units. For a perfectly symmetric distribution, the mean 
equals the median, and the skewness is zero. 

Extensive literature is available on the definition, use, and impli- 
cation of a, a, and other measures of the size distribution. (Inman, 
1952; Folk and Ward, 1957; McCammon, 1962; Folk, 1965, 1966; and 
Griffiths, 1967), but despite a long history of investigation and a 
considerable background of data, applications of size distribution infor- 
mation "are still largely empirical, qualitative, and open to alternative 
interpretations." (Griffiths, 1967, p. 104.) 

Currently, median grain size is the most commonly reported sand size 
characteristic, probably because there are only limited data to show the 
usefulness of other size distribution parameters in coastal engineering 
design. However, the standard deviation (Equation 4-4) must also be 
given as a parameter for use in beach fill design. (See Section 5,3; 
Krumbein and James, 1965; Vallianos, 1970; Berg and Duane, 1968.) 

4.212 Composition . In addition to classification by size, littoral 
material may be classified by composition. For shore protection pur- 
poses, composition normally is not an important factor, since the domi- 
nant littoral materials are quartz sands which are durable and chemically 
inert for periods longer than typical project lifetimes. However, sedi- 
ment composition is useful when the material departs from this expected 
condition. Other than quartz, littoral material may be composed of car- 
bonates (usually shell, coral, and algal material), organics (most often 
peat), and clays and silts (marsh and tidal flat deposits). 

4.213 Other Characteristics . In addition to size and composition, sedi- 
ments have a number of other properties by which they may be classified. 
Table 4-2 lists some density-related properties. Radioactive properties 
of naturally occurring thorium minerals have been used as tracers in beach 
sands. (Kamel, 1962; Kamel and Johnson, 1962, p. 324.) Other properties 
more directly related to soil mechanics studies are found in soil mechanics 
manuals. 

4.22 SAND AND GRAVEL 

By definition the word sand refers to a size class of material, but 
sand also implies the particular composition, usually quartz (silica). 



4-18 



Table 4-2. Density of Littoral Materials 



Specific Gravity (dimensionless) 


Quartz 


2.65 




Calcite 


2.72 




Heavy Minerals 


> 2.87 (commonly 2.87 to 3.33) 


Unit Weight* (Ibs./ft?) 


Uniform Sand 


Dry 


Saturated 






loose 


90 


118 


dense 


109 


130 


Mixed Sand 






loose 


99 


124 


dense 


116 


135 


Clay 






stiff glacial 





129 


soft, very organic 




89 



^From Terzaghi and Peck, 1967. 



In tropical climates, calcium carbonate, especially shell material, 
is often the dominant material in beach sand. In temperate climates, 
quartz and feldspar grains are the most abundant, commonly accounting for 
about 90 percent of beach sand. (Krumbein and Sloss, 1963, p. 134.) 

Because of its resistance to physical and chemical changes, and its 
common occurrence in terrestrial rocks, quartz is the most common mineral 
found in littoral materials. The relative abundance of non-quartz mate- 
rials is a function of the relative importance of the sources supplying 
the littoral zone and the materials available to those sources. The small 
amount of heavy minerals (specific gravity greater than 2.87) usually 
found in sand samples may indicate the source area of the material 
(McMaster, 1954; Giles and Pilkey, 1965; Judge, 1970), and thus they 
may be used as natural tracers. Such heavy minerals may form black or 
reddish concentrations at the base of dune scarps, along the berm, and 
around inlets. Occasionally heavy minerals occur in concentrations great 
enough to justify mining them as a metal ore. (Everts, 1971; Martens, 1928.) 
Table 4-3 from Pettijohn (1957, p. 117) lists the 26 most common minerals 
found in beach sands. 

Sand is by far the most important littoral material in coastal engi- 
neering design. However, in some localities, such as New England, Oregon, 
Washington, and countries bordering on the North Sea, gravel and shingle 
are locally important. Gravel-sized particles are often rock fragments, 
that is, a mixture of different minerals, whereas sand-sized particles 
usually consist of single mineral grains. 



4-19 



Table 4-3. Minerals Occurring in Beach Sand 



Common Dominant Constituents* 


Quartz— may average about 


Feldspar— typically only 


Calcite -includes shell, coral, 


70 percent in beach sand; 


10 to 20 percent in beach 


a%al fragments, and oolites; 


varies from near to over 


sands, but may be much 


varies from to nearly 100 


99 percent. 


more, particularly in 


percent; may include signifi- 




regions of eroding igneous 
rock. 


cant quantities of aragonite. 


Common Accessory Minerals (Adopted from 


Pettijohn, 1957) 


Andalusite Epidote 


♦Muscovite 


Apatite Garnet 


Rutile 


Aragonite Hornblende 


Sphene 


Augite Hypersthene-enstatite 


Staurolite 


Biotite llmenite 


Tourmaline 


Chlorite Kyanite 


Zircon 


Diopside Leucoxene 


Zoisite 


♦Dolomite Magnetite 





* These are light minerals with specific gravity not exceeding 2.87. The remaining 
minerals are heavy minerals with specific gravity greater than 2.87. Heavy minerals 
make up less than 1 percent of most beach sands. 



4.23 COHESIVE MATERIALS 

The amount of fine-grained, cohesive materials, such as clay, silt, 
and peat, in the littoral zone depends on the wave climate, contributions 
of fine sediment from rivers and other sources, and recent geologic his- 
tory. Fine-grain size material is common in the littoral zone wherever 
the annual mean breaker height is below about 1.0 foot. Fine material is 
found at or near the surface along the coasts of Georgia, western Florida 
between Tampa and Cape San Bias, and in large bays such as Chesapeake Bay 
and Long Island Sound. These are all areas of low mean breaker height. 
In contrast, fine sediment is seldom found along the Pacific coast of 
California, Oregon, and Washington, where annual mean breaker height 
usually exceeds 2.5 feet. 

Where rivers bring large quantities of sediment to the sea, the 
amount of fine material remaining along the coast depends on the balance 
between wave action acting to erode the fines and river deposition act- 
ing to replenish the fines. (Wright and Coleman, 1972.) The effect of 
the Mississippi River delta deposits on the coast of Louisiana is a pri- 
mary example. 

Along eroding, low-lying coasts, the sea moves inland over areas 
formerly protected by beaches, so that the present shoreline often lies 
where tidal flats, lagoons and marshes used to be. The littoral materials 
on such coasts may include silt, clay and organic material at shallow 
depths. As the active sand beach is pushed back, these former tidal flats 



4-20 



and marshes then outcrop along the shore, (e.g. Kraft, 1971.) Many barrier 
islands along the Atlantic and Gulf coasts contain tidal and marsh deposits 
at or near the surface of the littoral zone. The fine material is often 
bound together by the roots of marsh plants to form a cohesive deposit 
that may function for a time as beach protection. 

4.24 CONSOLIDATED MATERIAL 

Along some coasts, the principal littoral materials are consolidated 
materials, such as rock, beach rock, and coral, rather than unconsolidated 
sand. Such consolidated materials protect a coast and resist shoreline 
changes. 

4.241 Rock. Exposed rock along a shore indicates that the rate at which 
sand is supplied to the coast is less than the potential rate of sand 
transport by waves and currents. Reaction of a rocky shore to wave attack 
is determined by the structure, degree of lithification, and ground water 
characteristics of the exposed rock, and by the severity of the wave 
climate. Protection of eroding cliffs is a complex problem involving 
geology, rock mechanics, and coastal engineering. Two examples of these 
problems are the protection of the cliffs at Newport, Rhode Island (U.S. 
Army, Corps of Engineers, 1965) and at Gay Head, Martha's Vineyard, 
Massachusetts. (U.S. Army, Corps of Engineers, 1970.) 

Most rocky shorelines are remarkably stable, with individual rock 
masses identified in photos taken 50 years apart. (Shepard and Grant, 
1947.) 

4.242 Beach Rock . A layer of friable to well-lithified rock often occurs 
at or near the surface of beaches in tropical and subtropical climates. 
This material consists of local beach sediment cemented with calcium car- 
bonate, and it is commonly known as beach rock. Beach rock is important 
to coastal engineers because it provides added protection to the coast, 
greatly reducing the magnitude of beach changes (Tanner, 1960), and be- 
cause beach rock may affect construction activities. (Gonzales, 1970.) 

According to Bricker (1971), beach rock is formed when saline waters 
evaporate in beach sjinds, depositing calcium carbonate from solution. The 
present active formation of beach rock is limited to tropical coasts, such 
as the Florida Keys, but rock resembling beach rock is common at shallow 
depths along the east coast of Florida, on some Louisiana beaches, and re- 
lated deposits have been reported as far north as the Fraser River Delta 
in Canada. Comprehensive discussion of the subject is given in Bricker 
(1971) and Russell (1970). 

4.243 Organic Reefs . Organic reefs are wave-resistant structures reach- 
ing to about mean sea level that have been formed by calcium carbonate 
secreting organisms. The most common reef-building organisms are herma- 
typic corals and coralline algae. Reef- forming corals are usually re- 
stricted to areas having winter temperatures above about 18°C (Shepard, 
1963, p. 351), but coralline algae have a wider range. On U.S. 



4-21 



coastlines, active coral reefs are restricted to southern Florida, Hawaii, 
Virgin Islands and Puerto Rico. On Some of the Florida coast, reeflike 
structures are produced by sabellariid worms. (Kirtley, 1971.) Organic 
reefs stabilize the shoreline and sometimes affect navigation. 

4.25 OCCURRENCE OF LITTORAL MATERIALS ON U.S. COASTS 

Littoral materials on U.S. coasts vary from consolidated rock to 
clays, but sand with median diameters between 0.1 and 1.0 millimeters 
(3.3 and phi) is most abundant. General information on littoral mate- 
rials is in the reports of the U.S. Army Corps of Engineers' National 
Shoreline Study; information on certain specific geological studies is 
available in Shepard and Wanless (1971); and information on specific 
engineering projects is published in Congressional documents and is 
available in reports of the Corps of Engineers. 

4.251 Atlantic Coast . The New England coast is generally characterized 
by rock headlands separating short beaches of sand or gravel. Exceptions 
to this dominant condition are the sandy beaches in northeastern Massachu- 
setts, and along Cape Cod, Martha's Vineyard, and Nantucket. 

From the eastern tip of Long Island, New York, to the southern tip 
of Florida, the littoral materials are characteristically sand with median 
diameters in the range of 0.2 to 0.6 millimeter (2.3 to 0.7 phi). This 
material is mainly quartz sand. In Florida, the percentage of calcium 
carbonate in the sand tends to increase going south until, south of the 
Palm Beach area, the sand becomes predominantly calcium carbonate. Size 
distributions for the Atlantic coast, compiled from a number of sources, 
are shown in Figure 4-9. (Bash, 1972.) Fine sediments and organic sedi- 
ments are common minor constituents of the littoral materials on these 
coasts, especially in South Carolina and Georgia. Beach rock and coquina 
are common at shallow depths along the Atlantic coast of Florida. 

4.252 Gulf Coast . The Gulf of Mexico coasts of Florida, Alabama, and 
Mississippi are characterized by fine white sand beaches and by stretches 
of swamp. The swampy stretches are mainly in Florida, extending from 
Cape Sable to Cape Romano, and from Tarpon Springs to the Ochlockonee 
River. (Shepard and Wanless, 1971, p. 163.) 

The Louisiana coast is dominated by the influence of the Mississippi 
River which has deposited large amounts of fine sediment around the delta 
from which wave action has winnowed small quantities of sand. Tul" oC-q 
has been deposited along barrier beaches offshore of a deeply indented 
marshy coast. West of the delta is a 75-mile stretch of shelly sand 
beaches and beach ridges. 

The Texas coast is a continuation of the Louisiana coastal plain ex- 
tending about 80 miles to Galveston Bay; from there a series of long, wide 
barrier islands extends to the Mexican border. Littoral materials in this 
area are dominantly fine sand, with median diameters between 0.1 and 0.2 
millimeter (3.3 and 2.3 phi). 



4-22 



1.6 



1.4 



1.2 



0.8 



0.6 



0.4 



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rwi 



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-0.49 



-0.26 



0.0 



0.32 



0.64 



.32 



2.32 




E 

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E 
o 



NEW JERSEY 



DELMARVA NORTH CAROLINA 



0.8 



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0.32 



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1.32 



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1 


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-0.26 



0.0 



0.32 



0.64 



1.32 



2.32 




100 200 Miles 

Figure 4-9. Sand Size Distribution Along the U.S. Atlantic Coast 



4-23 



4.253 Pacific Coast . Sands on the southern California coast range in 
size from 0.1 to 0.6 millimeter (3.3 to 0.7 phi). (Emery, 1960, p. 190.) 
The northern California coast becomes increasingly rocky, and coarser 
material becomes more abundant. The Oregon and Washington coasts include 
considerable sand (Bascom, 1964), with many rock outcrops. Sand-sized 
sediment is contributed by the Columbia River and other smaller rivers. 

4.254 Alaska . Alaska has a long coastline (47,300 miles), and is corres- 
pondingly variable in littoral materials. However, beaches are generally 
narrow, steep, and coarse-grained; they commonly lie at the base of sea- 
cliffs. (Sellman, et al., 1971, p. D-10.) Quartz sand is less common and 
gravel more common here than on many other U.S. coasts. 

4.255 Hawaii . Much of the Hawaiian islands is bounded by steep cliffs, 
but there are extensive beaches. Littoral materials consist primarily of 
bed rock, and white sand formed from calcium carbonate produced by marine 
invertebrates. Dark colored basaltic and olivine sands are common where 
river mouths reach the sea. (Shepard and Wanless, 1971, p. 497, U.S. Army, 
Corps of Engineers, 1971.) 

4.256 Great Lakes . The U.S. coasts of the Great Lakes vary from high 
bluffs of clay, shale, and rock, through lower rocky shores and sandy 
beaches, to low marshy clay flats. (U.S. Army Corps of Engineers, National 
Shoreline Study, August 1971, North Central Division, p. 13.) The litto- 
ral materials are quite variable. Specific features are discussed, for 
example, by Bowman (1951); Hulsey (1962); Davis (1964-65); Bajorunas and 
Duane (1967); Berg and Duane (1968); Saylor and Upchurch (1970); Hands 
(1970); and Corps of Engineers (1953, 1971). 

4.26 SAMPLING LITTORAL MATERIALS 

Sampling programs are designed to provide information about littoral 
materials on one or more of the following characteristics: 

(a) typical grain size (usually median size), 

(b) size distribution, 

(c) composition of the littoral materials, 

(d) variation of (a), (b) , and (c) , with horizontal and vertical 
position on the site, and 

(e) possible variation in (a), (b) , (c) , and (d) with time. 

A sampling program will depend on the intended purpose of the samples, 
the time and money available for sampling, and an inspection of the site 
to be sampled. A brief inspection will often identify the principal vari- 
ations in the sediment and suggest the best ways to sample these variations. 
Sampling programs usually involve beach and nearshore sands and potential 
borrow sources. 



4-24 



The extent of sampling depends on the importance of littoral materials 
as related to the total engineering problem. The sampling program should 
specify: 

(a) horizontal location of sample, 

(b) spacing between samples, 

(c) volume of sample, 

(d) vertical location and type of sampled volume (e.g. surface layer 
or vertical core) , 

(e) technique for sampling, 

(f) method of storing and documenting the sample. 

Beaches typically show more variation across the profile than along 
the shore, so sampling to determine variation in the littoral zone should 
usually be along a line perpendicular to the shoreline. 

For reconnaissance sampling, a sample from both the wetted beach face 
and from the dunes is recommended. More extensive samples could be ob- 
tained at constant spacings across the beach or at different locations on 
the beach profile. Spacings between sampling lines are determined by the 
variation visible along the beach or by statistical techniques. 

Many beaches have subsurface layers of peat or other fine material. 
If this material will affect the engineering problem, vertical holes or 
borings should be made to obtain samples at depth. 

Sample volume should be adequate for analysis. For sieve analysis, 
about 50 grams are required; for settling tube analysis, smaller quanti- 
ties will suffice, but at least 50 grams are needed if other studies are 
required later. A quarter of a cup is more than adequate for most uses. 

Sand often occurs in fine laminae on beaches. However, for engineer- 
ing applications it is rarely necessary to sample individual laminae of 
sand. It is easier and more representative to take an equi dimensional 
sample that cuts across many laminae. Experience at CERC suggests that 
any method of obtaining an adequate volume of sample covering a few inches 
in depth usually gives satisfactory results. Cores should be taken where 
pile foundations are planned. 

The sample is only as good as the information identifying it. The 
following minimum information should be recorded at the time of sampling: 
locality, date and time, position on beach, remarks, and initials of col- 
lector. This information must stay with the sample, which is best ensured 
by fixing it to the sample container or placing it inside the container. 
Unless precautions are taken, the sample label may deteriorate due to 
moisture, abrasion, or other causes. Improved labels result by using 

4-25 



ballpoint ink on plastic strip (plastic orange flagging commonly used by- 
surveyors). Some information may be preprinted by rubber stamp on the 
plastic strip using indelible laundry ink. The advantage is that the 
label can be stored in the bag with the wet sample without the label 
deteriorating or the information washing or wearing off. 

4.27 SIZE ANALYSES 

Three common methods of analyzing a beach sediment for size are: 
visual comparison with a standard, sieve analysis, and settling tube 
analysis. 

The mean size of a sand sample can be estimated qualitatively by 
visually comparing the sample with sands of known sizes. Standards can 
be easily prepared by sieving out selected diameters, or by selecting 
samples whose sizes are already known. The standards may be kept in 
labeled treinsparent vials, or glued on cards. If glued, care is neces- 
sary to ensure that the particles retained by the glue are truly repre- 
sentative of the standard. 

Good, qualitative, visual estimates of mean size are possible with 
little previous experience. With experience, such visual estimates 
become semiquantitative. Visual comparison with a standard is a useful 
tool in reconnaisance, and in obtaining interim results pending a more 
complete laboratory size analysis. 

4.271 Sieve Analysis . Sieves are graduated in size of opening according 
to the U.S. Standard series. These standard sieve openings vary by a fac- 
tor of 1.19 from one opening to the next larger (by the fourth root of 2, 
or quarter phi intervals), e.g., 0.25, 0.30, 0.35, 0.42, and 0.50 milli- 
meters (2.00, 1.75, 1.50, 1.25, 1.00 phi). The range of sieve sizes used, 
and the size interval between sieves selected can be varied as required. 
Typical beach sand can be analyzed adequately using sieves with openings 
ranging from 0.062 to 2.0 millimeters (4.0 to -1.0 phi), in size incre- 
ments increasing by a factor of 1.41 (half-phi intervals). 

Sediment is usually sieved dry. However, for field analysis or for 
size analysis of sediment with a high content of fine material, it may be 
useful to wet-sieve the sediment. Such wet-sieve analyses are described 
by (e.g., Lee, Yancy, and Wilde, 1970, p. 4). 

Size analysis by sieves is relatively slow, but provides a widely 
accepted standard of reference. 

4.272 Settling Tube . Spherical sedimentary particles settle through 
water at a speed that increases as the particle weight increases. Since 
most sand is approximately spherical quartz, or calcium carbonate with a 
specific gravity near quartz, particle size is proportional to particle 
weight. Thus fall velocity can be used to measure size, (e.g., Colby and 
Christensen, 1956; Zeigler and Gill, 1959; and Gibbs, 1972.) Figure 4-31 
shows fall velocity for quartz spheres as a function of temperature. 



4-26 



There are numerous types of settling tubes; the most common is the 
visual accumulation tube (Colby and Christensen, 1956), of which there 
are also several types. The type now used at CERC (the rapid sediment 
analyzer or RSA) works in the following way: 

A 3- to 6-gram sample of sand is dropped through a tube filled with 
distilled water at constant temperature. A pressure sensor near the 
bottom of the tube senses the added weight of the sediment supported by 
the column of water above the sensor. As the sediment falls past the 
sensor, the pressure decreases. The record of pressure versus time is 
empirically calibrated to give size distribution based on fall velocity. 
(Zeigler and Gill, 1959.) 

The advantage of settling tube analysis is its speed. With modem 
settling tubes, average time for size analyses of bulk lots can be about 
one-fifth the time required by sieves. 

It is often claimed that a settling tube also provides a physically 
more realistic size analysis than a sieve, since the fall velocity takes 
into account the hydrodynamic effects of shape and density. However, 
this claim has not been documented, and may be questioned in view of the 
limited knowledge concerning the fluid mechanics of a sand sample falling 
in a settling tube - the lead particles encounter effectively laminar flow, 
the trailing particles encounter turbulent flow, and all particles inter- 
act with each other. 

Because of lack of an accepted standard settling tube, rapidly chang- 
ing technology, possible changes in tube calibration, and the uncertainty 
about fluid mechanics in settling tubes, it is recommended that all set- 
tling tubes be carefully calibrated by running a range of samples through 
both the settling tube and ASTM standard sieves. After thorough initial 
calibration, the calibration should be spot-checked periodically by running 
replicate sand samples of known size distribution through the tube. 

4.3 LITTORAL WAVE CONDITIONS 

4.31 EFFECT OF WAVE CONDITIONS ON SEDIMENT TRANSPORT 

Waves arriving at the shore are the primary cause of sediment trans- 
port in the littoral zone. Higher waves break further offshore, widening 
the surf zone and setting more sand in motion. Changes in wave period or 
height result in moving sand onshore or offshore. The angle between the 
crest of the breaking wave and the shoreline determines the direction of 
the longshore component of water motion in the surf zone, and usually the 
lon^chore transport direction. For these reasons, knowledge about the 
wave climate - the combined distribution of height, period, and direction 
through the seasons - is required for an adequate understanding of the 
littoral processes of any specific area. 



4-27 



4.32 FACTORS DETERMINING LITTORAL WAVE CLIMATE 

The wave climate at a shoreline depends on the offshore wave climate, 
caused by prevailing winds and storms, and on the bottom topography that 
modifies the waves as they travel shoreward. 

4.321 Offshore Wave Climate . Wave climate is the distribution of wave 
conditions averaged over the years. A wave condition is the particular 
combination of wave heights, wave periods, and wave directions at a given 
time. A specific wave condition offshore is the result of local winds 
blowing at the time of the observation and the recent history of winds 

in the more distant parts of the same water body. For local winds, wave 
conditions offshore depend on the wind velocity, duration, and fetch. 
For waves reaching an observation point from distant parts of the sea, 
a decay factor is added which preferentially filters out the higher, 
shorter period waves with increasing distances. (Chapter 3 discusses 
wave generation and decay.) 

4.322 Effect of Bottom Topography . As waves travel from deep water, they 
change height and direction because of refraction, shoaling, bottom fric- 
tion, and percolation. Laboratory experiments indicate that height and 
apparent period are also changed by nonlinear deformation of the waves in 
shallow water. 

Refraction is the bending of wave crests due to the slowing down of 
that part of the wave crest which is in shallower water. (See Section 
2.32.) As a result, refraction tends to decrease the angle between the 
wave crest and the bottom contour. Thus, for most coasts, refraction re- 
duces the breaker angle and spreads the wave energy over a longer crest 
length. 

Shoaling is the change in wave height due to conservation of energy 
flux. (See Section 2.32). As a wave moves into shallow water the wave 
height first decreases slightly, and then increases continuously to the 
breaker position, assuming friction and refraction effects are negligible. 

Bottom friction is important in reducing wave height where waves 
must travel long distances in shallow water. (Bretschneider, 1954.) 

There has been only limited field study of nonlinear deformation in 
shallow water waves (Byrne, 1969), but because such deformation is common 
in laboratory experiments (Calvin, 1972), it is expected that such phe- 
nomena are also common in the field. An effect of nonlinear deformation 
is to split the incoming wave crest into two or more crests affecting 
both the resulting wave height and the apparent period. 

Offshore islands, shoals, and other variations in hydrography also 
shelter parts of the shore. In general, bottom hydrography has the 
greatest influence on waves traveling long distances in shallow water. 
Because of the effects of bottom hydrography, nearshore waves generally 
have different characteristics than they had in deep water offshore. 



4-28 



Such differences are often visible on aerial photos. Photos may- 
show two or more distinct wave trains in the nearshore area, with the 
wave train most apparent offshore decreasing in importance as the surf 
zone is approached, (e.g., Harris, 1972.) The difference appears to be 
caused by the effects of refraction and shoaling on waves of different 
periods. Longer period waves, which may be only slightly visible off- 
shore, may become the most prominent waves at breaking, because shoaling 
increases their height relative to the shorter period waves. Thus, the 
wave period measured from the dominant wave offshore may be less than the 
wave period measured from the dominant wave entering the surf zone when 
two wave trains of unequal period reach the shore at the same time. 

4.323 Winds and Storms . The relation of a shoreline locality to the 
seasonal distribution of winds and to storm tracks is a major factor 
in determining the wave energy available for littoral transport. For 
example, strong winter winds in the northeastern United States usually 
are from the northwest, but because they blow from land to sea, they do 
not produce large waves at the shore. These northwest winds often imme- 
diately follow a northeaster - a low pressure system with strong north- 
east winds that generate high waves offshore. 

A storm near the coastline will influence wave climate with storm 
surge and high seas; a storm offshore will influence wave climate only 
by swell. The relation between the meteorological severity of a storm 
and the resulting beach change is complicated. (See Section 4.35.) 
Storms are not uniformly distributed in time or space: storms vary 
seasonally and from year to year; storms originate more frequently in 
some areas than in others; and storms follow characteristic tracks deter- 
mined by prevailing global circulation and weather patterns. 

An investigation of 170 damaging storms affecting the east coast of 
the U.S. from 1921-1962 (Mather, et al., 1964), classified the storms into 
eight types based on origin, structure, and path of movement. Of these 
eight types, although 33 percent were hurricanes, two types, comprising 
only 19 percent of the total, characterized by weather fronts east and 
south of the U.S. coasts produced more damage per storm because of long 
fetches. (Damage is defined by Mather, et al., as "at best some water 
damage," and includes "wave damage, coastal flooding, and tidal inunda- 
tion," but specificially excludes wind damage.) 

The probability that a given section of coast will experience storm 
waves depends on its ocean exposure, its location in relation to storm 
tracks, and the shelf bathymetry. 

4.33 NEARSHORE WAVE CLIMATE 

4.331 Mean Value Data on U.S. Littoral Wave Climates . Wave height and 
period data for some localities of the U.S. are becoming increasingly 
available (e.g., Thompson and Harris, 1972), but most localities still 
lack such data. However, wave direction is difficult to measure, and 
consequently direction data are rarely available. 



4-29 



The quality and quantity of available data often do not justify 
elaborate statistical analysis. Even where adequate data are available, 
a simple characterization of wave climate meets many engineering needs. 
While mean values of height and, to a lesser degree, period are useful, 
data on wave direction are generally of insufficient quality for even 
mean value use. Table 4-4 compiles mean annual wave heights collected 
from a number of wave gages and by visual observers along the coasts of 
the United States. Visual observations were made from the beach of waves 
near breaking. The gages were fixed in depths of 10 to 28 feet. 

Wave data treated in this section are limited to nearshore observa- 
tions and measurements. Consequently waves were fully refracted and had 
been fully affected by bottom friction, percolation, and nonlinear changes 
in wave form by shoaling. Thus, these data differ from data that would 
be obtained by simple shoaling calculations based on the deepwater wave 
statistics. In addition, data are normally lacking for the rarer, high- 
wave events. For these reasons, the data should not be used for struc- 
tural design, since they are only applicable to the particular site 
where they were obtained. Normal design practice is based on deepwater 
wave statistics which are then adjusted to the shallow-site conditions. 
However, the nearshore data are useful in littoral transport calculations. 

Mean wave height and period from a number of visual observations by 
coastguardmen at shore stations are plotted by month in Figures 4-10 and 
4-11, using the average values of stations within each of five coastal 
segments. Table 4-4 and Figures 4-10 and 4-11 show average values char- 
acteristic of the wave climate in exposed coastal localities. Visual 
height data represent an average value of the higher waves just before 
they first break. The data provide only approximate indications of the 
height distributions, but mean values of these distributions are useful. 

In Figure 4-10 the minimum monthly mean littoral zone wave height 
averaged for the California, Oregon, and Washington coasts exceeds the 
maxirmm mean littoral zone wave height averaged for the other coasts. 
This difference greatly affects the potential for sediment transport in 
the respective littoral zones, and should be considered by engineers 
when applying experience gained in a locality with one nearshore wave 
climate to a problem at a locality with another wave climate. 

4.332 Mean vs Extreme Conditions . Section 3.22 contains a discussion 
of wave height distributions and the relations between various wave 
height statistics, such as the mean, significant, and RMS heights, and 
extreme values. In general, a group of waves from the same record can be 
approximately described by a Rayleigh distribution. (See Section 3.22.) 
However, a different distribution appears necessary to describe the dis- 
tribution of significant wave heights, where each significant wave height 
is from a different wave record at the locality. (See Figure 4-12.) 

Visual analysis of waves recorded on chart paper is discussed in 
Section 3.22 and by Draper (1967), Tucker (1963), Harris (1970), and 
Magoon (1970). Spectrum analysis of wave records is discussed in 



4-30 




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c 



O 

•H 

> 



o 

to 

cd V) 

<u +J 

Z C 

<D 

i-H S) 

x; <u 

■l-> CO 

c 

o ■-• 

S rt 

§S 

0) o 



I 

u 

3 
M 



4-32 



Tabic 44. Mean Wave Height at Coaslal Localities of Conterminous United States 




Mean Aimual 




Mean Aiuuial 


Location 


Wave Height (ft.) 


Location 


Wave Height (ft.) 




Atlanl 


c Coast 




Maine 




New Jcrse) (cont.) 




Moose Peak 


1.5 


+ Lndlam Island 


1.9 


New Hampsliire 




Maryland 




liuni|ilon IVacli 


1.4 


()<<an ( jty 


1.8 


Massncliiisetts 




Virginia 




Nausel 


1.8 


"t" As.saleuipie 


2.6 


t Cape Cod 


2.5 


•Virginia Beach 


1.8 


Rhode Island 




Virginia Beach 


2.0 


Point Judith 


1.8 


North Carolina 




+ Mis((uamicut 


1.4 


•Nags Head 


3.0 


New York 




Nags Head 


3.9 


+ Southampton 


1.9 


-|-Wriglitsville 


2.3 


+ Westham|iton 


2.6 


Oak Island 


1.2 


"hjoiics Beach 


2.6 


tllolden Beach 


1.7 


Short Beacli 


1.7 


Georgia 




New Jersey 




St. Simon Island 


0.4 


Monmouth 


1.7 


Florida 




+ Deal 


2.3 


•Daylona Beach 


1.9 


Toms River 


2.0 


Ponce deLcon 


2.2 


+ Briganline 


2.2 


•Lake Worth 


2.3 


•Allanlic City 


2.8 


•Palm Beach 


2.3 


Atlantic City (BEP) 


1.3 


+ Boca Raton 


1.9 


tAtlanlicCity (CG) 


1.9 


llillsboro 


1.3 


GuH Coast 


Florida 




Florida (cont.) 




•Naples 


1.0 


^ INavarre Beach 


2.3 


Cape San Bias 


0.7 


Santa Rosa 


1.4 


^Panama City 


1.7 


Loiiisania 




^^Greyton Beach 


1.7 


Grand Island 


1.4 


^Crystal Beach 


1.7 


Texas 




+ Beasley Park 


1.8 


•Galveston 


1.4 


Pacific Coast 


California 




California (cont.) 




Point Loma 


2.1 


Point Argnello 


2.5 


+South Carlsbad 


2.7 


4= Natural Bridges 


2.9 


+ Carlsbad 


2.9 


+ Thornton 


4.1 


•Huntington Beacli 


1.7 


+ Goal Rock 


4.6 


^Huntington 


2.6 


Point Arena 


2.6 


+ Bolsa Chica 


2.2 


+ Prairie Creek 


3.7 


+ Leo Carrillo 


2.3 


Oregon 




^PEG at Point Mngii 


3.0 


Umpi[ua River 


3.3 


•Point Mugu 


2.7 


Yaiinina Bay 


5.1 


*:MeGralh 


3.5 


Washington 




^Carpinteria 


1.8 


Willapa Bay 


1.8 


Point Conception 


2.8 


Cape Flattery 


1.7 


^El Capitan 


2.0 







• CERC wave gage records. 

The following mean wave heights arc from visual (ncarbrcakcr) observacionj. 

t CERC Beach Evaluation Program. 

t CERC Littoral Environment Observation Program. 

Unmarked Coast Guard Observations. 



4-33 




o 
o 

O) 

CC 

i_ 
o 

CD 

>- 
I 



01 
O) 

en 
o 
O 

> 



to 
o 
o 
o 

e 

o 



X 
> 



o 
o 



c 
CO 



c 
o 



3 
J3 



to 
Q 



3 
C7> 



00 to 't CM 

i39^'il|6|3H 9ADM ;UD3!^|U5lS 



4-34 



Section 3.23 and by Kinsman (1965), National Academy of Sciences (1963), 
and Neumann and Pierson (1966). 

For the distribution of significant wave heights as defined by the 
data reduction procedures at CERC (Thompson and Harris, 1972), the data 
fit a modified exponential distribution of form 



fH>h.) = ^ 



"s "s min 



(4-6) 



where Hg is the significant height, Hg the significant height of in- 
terest, Hg Yrrin is the approximate "minimum significant height," and a 
is the significant wave height standard deviation. This equation depends 
on two parameters, Hg ^yi and a which are related to the mean height, 

H = H . + a . (4-7) 

I^ ^s min °^ ° ^^® "°^ available, but the mean significant height, 
Hg is known, then an approximation to the distribution of (4-6) can be 
obtained from the data of Thompson and Harris (1972, Table 1), which 
suggest 

H.^.-„ - 0.38 H^ . (4-8) 

This approximation reduces Equation 4-6 to a one-parameter distribution 
depending only on mean significant wave height 



f(h^>h> 



1.61 H —0.61 H 

s s 



". 



(4-9) 



Equation 4-9 is not a substitute for the complete distribution function, 
but when used with the wave-gage data on Figure 4-12, it provides an 
estimate of higher waves with agreement within 20 percent. Greater 
scatter would be expected with visual observations. 

4.34 OFFICE STUDY OF WAVE CLIMATE 

Information on wave climate is necessary for understanding local 
littoral processes. Usually, time does not permit obtaining data from 
the field, and it is necessary to compile information in an office study. 
The primary variables of engineering interest for such a compilation are 
wave height and direction. 

Shipboard observations covering conterminous U.S. coasts and other 
ocean areas are available as summaries (Summary of Synoptic Meterological 
Observations, SSMO) through the National Technical Information Service, 
Springfield, Va. 22151. See Harris (1972) for a preliminary evaluation 
of this data for coastal engineering use. 



4-35 



When data are not available for a specific beach, the wave climate 
can be estimated by extrapolating from another location, after correcting 
for differences in coastal exposure, winds, and storms. 

On the east, gulf, and Great Lakes coasts, local winds are often 
highly correlated with the direction of longshore currents. Such wind 
data are available in "Local Climatological Data" sheets published monthly 
by the National Weather Service, National Oceanographic and Atmospheric 
Agency (NOAA) for about 300 U.S. weather stations. Other NOAA wind-data 
sources include annual summaries of the Local Climatological Data by sta- 
tion (Local Climatological Data with Comparative Data), and weekly sum- 
maries of the observed weather (Daily Weather Maps), all of which can be 
ordered from the Superintendent of Documents, U.S. Government Printing 
Office, Washington, D.C. 20402. 

Local weather data are often affected by conditions in the neighbor- 
hood of the weather station, so care should be used in extrapolating 
weather records from inland stations to a coastal locality. However, 
statistics on frequency and severity of storm conditions do not change 
appreciably for long reaches of the coast. For example, in a study of 
Texas hurricanes, Bodine (1969) felt justified in assuming no difference 
in hurricane frequency along the Texas coast. In developing information 
on the Standard Project Hurricane, Graham and Nunn (1959) divided the 
Atlantic Coast into zones 200 miles long and the gulf coast into zones 
400 miles long. Variation of most hurricane parameters within zones is 
not great along straight open stretches of coast. 

The use of weather charts for wave hindcasting is discussed in Sec- 
tion 3.4. Computer methods for generating offshore wave climate are now 
(1973) under test and development. However, development of nearshore 
wave climate from hindcasting is usually a time-consuming job, and the 
estimate obtained may suffer in quality because of the inaccuracy of 
hindcast data, and the difficulty of assessing the effect of nearshore 
topography on wave statistics. At the present time, if available at the 
specific location, statistics based on wave-gage records are preferable 
to hindcast statistics when wave data for the shallow-water conditions 
are required. 

Other possible sources of wave climate information for office stud- 
ies include aerial photography, newspaper records, and comments from 
local residents. 

Data of greater detail and reliability than that obtained in an 
office study can be obtained by recording the wave conditions at the 
shoreline locality for at least 1 year by the use of visual observers 
or wave gages. A study of year-to-year variation in wave height statis- 
tics collected at CERC wave gages (Thompson and Harris, 1972), indicates 
that six observations per day for 1 year gives a reliable wave height 
distribution function to the 1 percent level of occurrence. At the gage 
at Atlantic City, one observation a day for 1 year provided a useful 
height-distribution function. 

4-36 



4.35 EFFECT OF EXTREME EVENTS 

Infrequent events of great magnitude, such as hurricanes, cause sig- 
nificant modification of the littoral zone, particularly to the profile 
of a beach. An extreme event could be defined as an event, great in 
terms of total energy expended or work done, that is not expected to occur 
at a particular location, on the average, more than once every 50 to 100 
years. Hurricane Camille in 1969 and the East Coast Storm of March 1962 
can be considered extreme events. Because large storms are infrequent, 
and because it does not necessarily follow that the magnitude of a storm 
determines the amount of geomorphic change, the relative importance of 
extreme events is difficult to establish. 

Wolman and Miller (1960) suggested that the equilibrium profile of 
a beach is more related to moderately strong winds that generate moderate 
storm waves, rather than to winds that accompany infrequent catastrophic 
events. Saville (1950) showed that for laboratory tests with constant 
wave energy and angle of attack there is a particular critical wave steep- 
ness at which littoral transport is a maximum. Under field conditions, 
there is probably a similar critical value that produces transport out of 
proportion to its frequency of occurrence. The winds associated with this 
critical wave steepness may be winds generated by smaller storms, rather 
than the winds associated with extreme events. 

A review of studies of beach changes caused by major storms indi- 
cates that no general conclusion that can be made concerning the signifi- 
cance of extreme events. Many variables affect the amount of damage a 
beach will sustain in a given storm. 

Most storms move large amounts of sand from the beach offshore, but 
after the storm, the lower waves that follow tend to restore this sand to 
the face of the beach. Depending on the extent of restoration, the storm 
may result in little permanent change. Depending on the path of the storm 
and the angle of the waves, a significant amount of material can also be 
moved alongshore. If the direction of longshore transport caused by the 
storm is opposite to the net direction of transport, the sand will prob- 
ably be returned in the months after the storm and permanent beach changes 
effected by the storm will be small. If the direction of transport before, 
during, and after the storm is the same, then large amounts of material 
could be moved by the storm with little possibility of restoration. Suc- 
cessive storms on the same beach may cause significant transport in oppo- 
site directions, (e.g. Everts, 1973.) 

There are some unique events that are only accomplished by catastro- 
phic storms. The combination of storm surge and high waves allows water 
to reach some areas not ordinarily attacked by waves. These extreme con- 
ditions may result in the overtopping of dunes and in the formation of 
washover fans and inlets. (Morgan, et al., 1958; Nichols and Marston, 
1939.) Some inlets are periodically reopened by storms and then sealed 
by littoral drift tremsported by normal wave action. 



4-37 



The wave climate at a particular beach also determines the effect 
a storm will have. In a high-energy climate, storm waves are not much 
larger than ordinary waves, and their effects may not be significant. An 
example of this might be northeasters occurring at Cape Cod. In a low- 
energy wave climate, where transport volumes are usually low, storm waves 
can move significant amounts of sand, as do hurricanes on the gulf coast. 

The type of beach sediment is also important in storm-induced changes. 
A storm can uncover sediments not ordinarily exposed to wave action, and 
thus alter the processes that follow the storm. (Morgan, et al., 1958.) 
In sand-deficient areas where the beach is underlain by mud, the effects 
of a storm can be severe and permanent. 

The effects of particular storms on certain beaches are described in 
the following paragraphs. These examples illustrate how an extreme event 
may affect the beach. 

In October 1963, the worst storm in the memory of the Eskimo people 
occurred over an ice-free part of the Arctic Ocean, and attacked the coast 
near Barrow, Alaska. (Hume and Schalk, 1967.) Detailed measurements of 
some of the key coastal areas had been made just before the storm. Freeze- 
up just after the storm preserved the changes to the beach until surveys 
could be made the following July. Most of the beaches accreted 1 to 2 
feet, although Point Barrow was turned into an island. According to Hume 
and Schalk, "The storm of 1963 would appear to have added to the Point the 
sediment of at least 20 years of normal longshore transport." Because of 
the low-energy wave climate and the short season in which littoral pro- 
cesses can occur at Barrow, this storm significantly modified the beach. 

A study of two hurricanes, Carla (1961) and Cindy (1963), was made by 
Hayes (1967). He concluded that "the importance of catastrophic storms 
as sediment movers cannot be over-emphasized," and observed that, in low- 
energy wave climates, most of the total energy is expended in the near- 
shore zone as a series of catastrophes. In this region, however, the rare 
"extreme" hurricane is probably not as significant in making net changes 
as the more frequent moderate hurricanes. 

Surprisingly, Hurricane Camille, with mEiximum winds of 200 mph, did 
not cause significant changes to the beaches of Mississippi and Louisiana. 
Tanner (1970) estimated that the sand transport along the beach appeared 
to have been an amount equal to less than a year's amount under ordinary 
conditions, and theorized that "the particular configuration of beach, sea 
wall, and coastal ridge tended to suppress large scale transport." 

Hurricane Audrey struck the western coast of Louisiana in June, 1957. 
The changes to the beach during the storm were not extreme nor permanent. 
However, the storm exposed marsh sediments in areas where sand was defi- 
cient, and "set the stage for a period of rapid shoreline retreat follow- 
ing the storm." (Morgan, et al., 1958.) Indirectly, then, the storm was 
responsible for significant geomorphic change. 



4-38 



A hurricane (unnamed) coincided with spring tide on the New England 
coast on 21 September 1938. Property damage and loss of life were both 
high. A storm of this magnitude was estimated to occur about once every 
150 years. A study of the beach changes along a 12-mile section of the 
Rhode Island coast (Nichol and Marsten, 1939) showed that most of the 
changes in the beach profile were temporary. The net result was some 
cliff erosion and a slight retrogression of the beaches. 

Beach changes from Hurricane Donna which hit Florida in September 
1960 were more severe and permanent. In a study of the southwestern 
coast of Florida before and after the storm. Tanner (1961), concluded 
that "Hurricane Donna appears to have done 100 year's work, considering 
the typical energy level thought to prevail in the area." 

On 1 April 1946, a tsunami struck the Hawaiian Islands with runup in 
places as high as 55 feet above sea level. (Shepard, et al., 1950.) The 
beach changes were similar to those inflicted by storm waves although "in 
only a few places were the changes greater than those produced during nor- 
mal storm seasons or even by single severe storms." Because a tsunami is 
of short duration, extensive beach changes do not occur, although property 
damage can be quite high. 

Several conclusions can be drawn from the above examples. If a beach 
has a sufficient sand supply and fairly high dunes, and if the dunes are 
not breached, little permanent modification will result from storms, except 
for a brief acceleration of the normal littoral processes. This accelera- 
tion will be more pronounced on a shore with low-energy wave conditions. 

4.4 NEARSHORE CURRENTS 

Nearshore currents in the littoral zone are predominantly wave-induced 
motions superimposed on the wave-induced oscillatory motion of the water. 
The net motions generally have low velocities, but because they transport 
whatever sand is set in motion by the wave-induced water motions, they are 
important in determining littoral transport. 

There is only slight exchange of fluid between the offshore and the 
surf zone. Onshore -offshore flows take place in a number of ways, which 
at present are not fully understood. 

4.41 WAVE- INDUCED WATER MOTION 

In idealized deepwater waves, water particles have a circular motion 
in a vertical plane perpendicular to the wave crest (Fig. 2-4, Section 
2.235), but this motion does not reach deep enough to affect sediment on 
the bottom. In depths where waves are affected by the bottom, the circu- 
lar motion becomes elliptical, and the water at the bottom begins to move. 
In shallow water, the ellipses elongate into nearly straight lines. At 
breaking, particle motion becomes more complicated, but even in the surf 
zone, the water moves forward and backward in paths that are mostly hori- 
zontal, with brief, but intense, vertical motions produced by the passage 

4-39 



of the breaker crest. Since it is this wave-induced water particle motion 
that causes the sediment to move, it is useful to know the length of the 
elliptical path traveled by the water particles and the maximum velocity 
and acceleration attained during this orbit. 

The basic equations for water-wave motion before breaking are dis- 
cussed in Chapter 2. Quantitative estimates of water motion are possible 
from small -amplitude wave theory (Section 2.23), even near breaking where 
assiomptions of the theory are not valid. (Dean, 1970; Eagleson, 1956.) 
Equations 2-13 and 2-14, in Section 2.234 give the fluid-particle velocity 
components u, w in a wave where small -amplitude theory is applicable. 
(See Figure 2-3 for relation to wave phase and water particle accelera- 
tion.) 

For sediment transport, the conditions of most interest are those 
when the wave is in shallow water. For this condition, and making the 
small-amplitude assumption, the horizontal length 2A, of the path moved 
by the water particle as a wave passes in shallow water is approximately 



HT^gd 
2A = -^^^ , C4-10) 



and the maximum horizontal water velocity is 

2d 



"^max = -TTT- ' (4-11) 



The term under the radical is the wave speed in shallow water. 
************** EXAMPLE PROBLEM *************** 

GIVEN : A wave 1 foot high with a period of 5 seconds is progressing shore- 
ward in a depth of 2 feet. 

FIND ; 

(a) Calculate the maximum horizontal distance 2A the water particle 
moves during the passing of a wave. 

(b) Determine the maximum horizontal velocity ^cuc °^ ^ water 
particle. 

(c) Compare the maximum horizontal distance 2A with the wavelength 
in the 2- foot depth. 

(d) Compare the maximum horizontal velocity ^cuc» with the wave 
speed, C. 



4-40 



solution: 



(a) Using Equation 4-10, the maximum horizontal distance is 

HT Vgd^ 



2A = 



2ffd 



1 (5) ^32.2 (2)^ 

2A = = 3.2 feet . 

27r(2) 

(b) Using Equation 4-11, the maximum horizontal velocity is 



2d 
1^32.2(2)^ 
2(2) 



u = TTT; = 2.0 feet per second 

max T '^^^ ^ 



(c) Using the relation L = Ti/gd to determine the shallow-water 
wavelength. 



L = 5v/32.2(2)' = 40.1 feet . 

From (a) above, the maximiam horizontal distance 2A is 3.2 feet 
therefore the ratio 2A/L is 

2A 3.2 

— = = 0.08 . 

L 40.1 

(d) Using the relation C = v'^ (Equation 2-9) to determine the 
shallow-water wave speed 



C = y^32.2 (2)" =8.0 feet per second . 

From (b) above the maximum horizontal velocity ^ax> ^^ ^'^ 
feet per second. Therefore the ratio ^ax/^ i^ 

^max 2.0 

-^^^ = = 0.25. 

C 8.0 

************************************* 

Although small -amplitude theory gives a fair understanding of many 
wave-related phenomena, there are important phenomena that it does not 
predict. Observation and a more complete analysis of wave motion show 
that particle orbits are not closed. Instead, the water particles 
advance a little in the direction of the wave motion each time the wave 



4-41 



passes. The rate of this advance is the mass transport vetoaity; (Equa- 
tion 2-55, Section 2.253). This velocity becomes important for sediment 
transport, especially for sediment suspended above ripples seaward of the 
breaker. 

For conditions evaluated at the bottom (z = -d) , the maximum bottom 
velocity, umoxr. j') . given by Equation 2-13 determines the average bottom 
mass transport velocity, u(_j). obtained from Equation 2-55, according to 
the equation 

("wax/ J)) 
"(-d)='-^' . (4-12) 

where C is the wave speed given by Equation 2-3. Equation 2-55, and 
thus Equation 4-12, does not include allowance for return flow which 
must be present to balance the mass transported in the direction of wave 
travel. In addition, the actual distribution of the time-averaged net 
velocity depends sensitively on such external factors as bottom character- 
istics, temperature distribution, and wind velocity. (Mei, Liu, and Carter, 
1972.) Most observations show the time-averaged net velocity near the 
bottom is directed toward the breaker region from both sides. (See Inman 
and Quinn, (1952), for field measurements in surf zone; Galvin and Eagle- 
son, (1965) for laboratory observations; and Mei, Liu and Carter (1972, 
p. 220), for comprehensive discussion.) However, both field and labora- 
tory observations have shown that wind-induced bottom currents may be 
great enough to reverse the direction of the shoreward time-averaged 
wave-induced velocity at the bottom when there are strong onshore winds. 
(Cook and Gorsline, 1972; and Kraai, 1969.) 

4.42 FLUID MOTION IN BREAKING WAVES . 

During most of the wave cycle in shallow water, the particle velo- 
city is approximately horizontal and constant over the depth, although 
right at breaking there are significant vertical velocities as the water 
is drawn up into the crest of the breaker. The maximum particle velocity 
under a breaking wave is approximated by solitary wave theory (Equation 
2-66) to be 

Hmax = '^ = v/g (H + d)' , (4-13) 

where (H+d) is the distance measured from crest of the breaker to the 
bottom. 

Fluid motions at breaking cause most of the sediment transport in 
the littoral zone, because the bottom velocities and turbulence at break- 
ing suspend more bottom sediment. This suspended sediment can then be 
transported by currents in the surf zone whose velocities are normally 
too low to move sediment at rest on the bottom. 

Tlie mode of breaking may vary significantly from spilling to plung- 
ing to collapsing to surging, as the beach slope increases or the wave 



4-42 



steepness (height-to-length ratio) decreases. (Galvin, 1967.) Of the 
four breaker types, spilling breakers most closely resemble the solitary 
waves whose speed is described by Equation 4-13. (Galvin, 1972.) Spill- 
ing breakers differ little in fluid motion from unbroken waves (Divoky, 
LeMehaute, and Lin, 1970), and thus tend to be less effective in trans- 
porting sediment than plunging or collapsing breakers. 

The most intense local fluid motions are produced by plunging break- 
ers. As the wave moves into shallower depths, the front face begins to 
steepen. When the wave reaches a mean depth about equal to its height, 
it breaks by curling over at the crest. The crest of the wave acts as a 
free-falling jet that scours a trough into the bottom. At the same time, 
just seaward of the trough, the longshore bar is formed, in part by sedi- 
ment scoured from the trough and in part by sediment transported in rip- 
ples moving from the offshore. 

The effect of the tide on nearshore currents is not discussed, but 
tide-generated currents may be superimposed on wave-generated nearshore 
currents, especially near estuaries. In addition, the changing elevation 
of the water level as the tide rises and falls may change the area and 
the shape of the profile through the surf zone, and thus alter the near- 
shore currents. 

4.43 ONSHORE-OFFSHORE CURRENTS 

4.431 Onshore-Offshore Exchange . Field and laboratory data indicate 
that water in the nearshore zone is divided by the breaker line into 
two distinct water masses between which there is only a limited exchange 
of water. 

The mechanisms for the exchange are: mass transport velocity in 
shoaling waves, wind-induced surface drift, wave-induced setup, currents 
induced by irregularities on the bottom, rip currents, and density cur- 
rents. The resulting flows are significantly influenced by, and act on, 
the hydrography of the surf and nearshore zones. Figure 4-13 shows the 
nearshore current system measured for particular wave conditions on the 
southern California coast. 

At first observation, there appears to be extensive exchange of 
water between the nearshore and the surf zone. However, the breaking 
wave itself is formed largely of water that has been withdrawn from the 
surf zone after breaking. (Galvin, 1967.) This water then reenters the 
surf zone as part of the new breaking wave, so that only a limited amount 
of water is actually transferred offshore. This inference is supported 
by the calculations of Longuet-Higgins (1970, p. 6788) which show that 
little mixing is needed to account for observed velocity distributions. 
Most of the exchange mechanisms indicated act with speeds much slower 
than the breaking-wave speed, which may be taken as an estimate of the 
maximum water particle speed in the littoral zone indicated by Equation 
4.13. 



4-43 



Scripps Canyon 



Hb 



2 December 1948 

Wave Period 15 Seconds 

Wave from WNW 

-.25 
.25-.50KN 
.50- 1.0 KN 
>l Knot 

Observed Current (not measured) 
Starting Position of Surface Float 
= Breaker Height 
Float Recovery Area 



— Hh = 4.5 



Scripps 
stitution 




k 



n 



<^y\/^---^ 



s^^" 



SCALE IN FEET 

500 1000 

t-l l-l i-l t-l l-l I 



(from Shepard and Inman, 1950) 



Figure 4-13. Nearshore Current System Near La Jolla Canyon, California 



4-44 



4.432 Diffuse Return Flow . Wind- and wave-induced water drift, pres- 
sure gradients at the bottom due to setup, density differences due to 
suspended sediment and temperature, and other mechanisms produce patterns 
of motion in the surf zone that vary from highly organized rip currents 

to broad diffuse flows that require continued observation to detect. Dif- 
fuse return flows may be visible in aerial photos as fronts of turbid 
water moving seaward from the surf zone. Such flows may be seen in the 
photos reproduced in Sonu (1972, p. 3239). 

4.433 Rip Currents . Most noticeable of the exchange mechanisms between 
offshore and surf zone are rip currents. (See Figure 4-14, and Figure 
A-7, Appendix A.) Rip currents are concentrated jets that carry water 
seaward through the breaker zone. They appear most noticeable when long, 
high waves produce wave setup on the beach. In addition to the classi- 
cal rip currents, there are other localized currents directed seaward 
from the shore. Some are due to concentrated flows down gullies in the 
beach face, and others can be attributed to interacting waves and edge 
wave phenomena. (Inman, Tait, and Nordstrom, 1971, p. 3493.) The origin 
of rip currents is discussed by Arthur (1962), and Sonu (1972). 

Three-dimensional circulation in the surf is documented by Shepard and 
Inman (1950), and this complex flow needs to be considered, especially in 
evaluating the results of laboratory tests for coastal engineering purposes. 
However, at present, there is no proven way to predict the conditions that 
produce rip currents or the spacing between rips. In addition, data are 
lacking that would indicate quantitatively how important rip currents are 
as sediment transporting agents. 

4.44 LONGSHORE CURRENTS 

4.441 Velocity and Flow Rate . Longshore currents flow parallel to the 
shoreline, and are restricted mainly between the zone of breaking waves 
and the shoreline. Most longshore currents are generated by the long- 
shore component of motion in waves that obliquely approach the shoreline. 

Longshore currents typically have mean values of 1 foot per second 
or less. Figure 4-15 shows a histogram of 5,591 longshore current veloc- 
ities measured at 36 sites in California during 1968. Despite frequent 
reports of exceptional longshore current speeds, most data agree with 
Figure 4-15 in showing that speeds above 3 feet per second are unusual. 
A compilation of 352 longshore current observations, most of which appear 
to be biased toward conditions producing high speed, showed that the maxi- 
mum observed speed was 5.5 feet per second, and that the highest observa- 
tions were reported to have been wind-aided. (Calvin and Nelson, 1967.) 
Although longshore currents generally have low speeds, they are important 
in littoral processes because they flow along the shore for extended peri- 
ods of time, transporting sediment set in motion by the breaking waves. 

The most important variable in determing the longshore current veloc- 
ity is the angle between the wave crest and the shoreline. However, the 
volume rate of flow of the current and the longshore transport rate depend 

4-45 




Figure 4-14. Typical Rip Currents, Ludlam 
Island, New Jersey 



4-46 



mostly on breaker height. The outer edge of the surf zone is determined 
by the breaker position. Since waves break in water depths approximately 
proportional to wave height, the width of the surf zone on a beach in- 
creases with wave height. This increase in width increases the cross 
section of the surf zone. 



2400 



2000 



10 

c 
o 



^ 1600 



o 

> 

w 

O 



Xi 

E 



1200 



800 



400 



T 



T 



T 



Total of 5591 Observations 
March -December 1968 



NORTH 



SOUTH 







Figure 4-15. 



ij -3 -2 -I 1 2 3 4 5 
Longshore Current Velocity, (feet per sec) 

Distribution of Longshore Current Velocities. Data taken 

from CERC California LEO Study (See Szuwalski 1970). 



If the surf zone cross section is approximated by a triangle, then 
an increase in height increases the area (and thus the volume of the flow) 
as the square of the height, which nearly offsets the increase in energy 
flux (which increases as the 5/2 power of height). Thus, the height is 
important in determining the width and volume rate of longshore current 
flow in the surf zone. (Calvin, 1972.) 

Longshore current velocity varies both across the surf zone (Longuet- 
Higgins, 1970b) and in the longshore direction (Calvin and Eagleson, 1965). 
Where an obstacle to the flow, such as a groin, extends through the surf 
zone, the longshore current speed downdrift of the obstacle is low, but 
it increases with distance downdrift. Laboratory data suggest that the 
current takes a longshore distance of about 10 surf widths to become fully 
developed. These same experiments (Calvin and Eagleson, 1965) suggest that 
the velocity profile varies more across the surf zone at the start of the 
flow than it does downdrift where the flow has fully developed. The ratio 
of longshore current speed at the breaker position to longshore current 
speed averaged across the surf zone varied from about 0.4 where the flow 
started to about 0.8 or 1.0 where the flow was fully developed. 



4-47 



4.442 Velocity Prediction . The variation in longshore current velocity 
across the surf zone and along the shore, and the uncertainties in vari- 
ables such as the surf zone hydrography, make prediction of longshore 
current velocity uncertain. There are three equations of possible use 
in predicting longshore currents: Longuet-Higgins (1970); an adaptation 
from Bruun (1963); and Galvin (1963). All three equations require co- 
efficients identified by comparing measured and computed velocities, and 
all three show about the same degree of agreement with data. Two sets of 
data (Putnam, et al., 1949, field data; Galvin and Eagleson, 1965, labora- 
tory data) appear to be the most appropriate for checking predictions. 

The radiation stress theory of Longuet-Higgins (1970a, Equation 62), 
as modified by fitting it to the data, is the one recommended for use 
based on its theoretical foundation. The other two semiempirical equa- 
tions may provide a check on the Longuet-Higgins prediction. Written 
in common symbols (m is beach slope; g is acceleration of gravity; H^ 
is breaker height; T is wave period; and a^j is angle between breaker 
crest and shoreline), these equations are: 

a. Longuet-Higgins . 

vj, = M, m(gHj,)*/^ sin2aj, , (4-14) 

where 



0.694 r(2i3) 



'2 



M, = / "^ (4-15) 

y 

According to Longuet-Higgins (1970a, p. 6788) , v^ is the longshore cur- 
rent speed at the breaker position, r is a mixing coefficient which 
ranges between 0.17 (little mixing) and 0.5 (complete mixing), but is 
commonly about 0.2; g is the depth-to-height ratio of breaking waves 
in shallow water taken to be 1.2 and fj? is the friction coefficient, 
taken to be 0.01. Using these values, M,= 9.0. 

Applying equation 4-14 to the two sets of data yields predictions 
that average about 0.43 of the measured values. In part, these predicted 
speeds are lower because v^j, as given in Equation 4-14 is for the speed 
at the breaker line, whereas the measured velocities are mostly from the 
faster zone of flow shoreward of the breaker line. (Galvin and Eagleson, 
1965.) Therefore, Equation 4-14 multiplied by 2.3 leads to the modified 
Longuet-Higgins equation for longshore current velocity: 

V = 20.7 m(gHj,)'/^ sin 2ay , (4-16) 

used in Figure 4-16. Further developments in the Longuet-Higgins' (1970b 
and 1971) theory permit calculation of velocity distribution, but there 
is no experience with these predictions for longshore currents flowing 
on erodible sand beds. 



4-48 



o 
a> 



T3 

a> 
a> 

o. 
CO 



3 



o 



C7> 



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0) 



CO 
O 

a> 



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XX — I 




z 




z 




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t z 




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=: ^ 


_± _ __ 7 - 


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IT Z 


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\-i' - Ar- -,Z 




t-<. dir 2 :^ - 


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tns^^rrr^tj z 








^2 




Z 




^2 


4 __ _ _ 


_ _ _ 7 _ _ __. 


H 


A ^ 




ft z 




^2 


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z 




^z 




z 


... ... ^^ 


^ T 




Z II 


■3 _,^ _ _ 


Z JI _ IV - - -- 


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f ^^ ,^ 2 




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f 


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aI , 




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a^L ' 


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-■ -.flli'i ri/tn C'l fnnrr llil iikK n n rl 


^<^ 




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i:^M5oLb]^2"«L 


z 




n I 


__— ^_-±± : ±.±: 







2 3 4 5 

Computed Longshore Current Speed (ft/sec) 



Figure 4-16. Measured Versus Predicted Longstiore Current Speed 



4-49 



b. Bruun (1963 as Modified) . 

^b = ^2 (^^bf ["'^b (si" 2aj,)/T]'^^ , (4-17) 

where v^ is the mean velocity in the surf zone where the flow is fully 
developed, and M2 involves a friction factor of the Chezy kind (see 
Galvin, 1967, p. 297.) 

c. Galvin (1963) . 

vj, = KgmTsin2aj, , (4-18) 

where v^ is the mean velocity in the surf zone where the flow is fully 
developed, and K is a coefficient depending on breaker height-to-depth 
ratio and the ratio of trough depression on breaker height. To a good 
approximation, K may be taken as 1.0. (Galvin and Eagleson, 1965.) 

4.45 SUMMARY 

The major currents in the littoral zone are wave-induced motions 
superimposed on the wave-induced oscillatory motion of the water. The 
net motions generally have low velocities, but because they transport 
whatever sand is set in motion by the wave-induced water motions, they 
are important in determining littoral transport. 

Evidence indicates that there is only slight exchange of fluid 
between the offshore and the surf zone. 

Longshore current velocities are most sensitive to changes in breaker 
angle, and to a lesser degree, to changes in breaker height. However, the 
volume rate of flow of the longshore current is most sensitive to breaker 
height, probably proportional to H^. The modified Longuet-Higgins equation 
(4-16) is recommended for predicting mean longshore current velocity of 
fully developed flows, and the two semiempirical equations (4-17 and 4-18) 
are available as checks on the Longuet-Higgins equation. 

4.5 LITTORAL TRANSPORT 

4.51 INTRODUCTION 

4.511 Importance of Littoral Transport . Sediment motions indicated by 
the shoreline configuration in Figure 4-17 are aspects of littoral trans- 
port. If the coast is examined on satellite imagery as shown in Figure 
4-17, only its general characteristics are visible. At this elevation, 
the shore consists of bright segments that are straight or slightly curved. 
The brightness is evidence of sand, the most common material along the 
shore. The straightness often is evidence of sediment transport. 

In places, the straight segments of shoreline cut across preexisting 
topography. Elsewhere, the shoreline segments are separated by wide la- 
goons from the irregular mainland. The fact that the shore is nearly 
straight across both mainland and irregular bays is evidence of headland 



4-50 





■< 'i. 


ATLANTIC 


V 




OCEAN 


•t' 




y 


Jones. Bea, 




\ 


Long.lsliJft^ ; 


New Y-ai 


jir 






g^j 


(New York Bight) 


" . *.t-*>*l|» . -'« 


V, 


i 




t 


^ 




\ 

Sandy Hook, 




New Jersey ' 


C' 


> ;», 




' '* •- "' 


^ 


'"■ "i 




New Jersey , a . 


■ 


mZ* ■ 




' .. ^ ^ ■• ■." 


1 


^^^^^ 




r\ 



Figure 4-17. Coasts in Vicinity of New York Bight 



4-5 



erosion, accompanied by longshore transport which has carried sand along 
the coast to supply the barriers and spits extending across the bays. 
The primary agent producing this erosion and transport is the action of 
waves impinging on the shore. 

Littoral transport is the movement of sedimentary material in the 
littoral zone by waves and currents. The littoral zone extends from the 
shoreline to just beyond the most seaward breakers. 

Littoral transport is classified as onshore-offshore transport or as 
longshore transport. Onshore-offshore transport has an average net direc- 
tion perpendicular to the shoreline; longshore transport has an average 
net direction parallel to the shoreline. The instantaneous motion of 
sedimentary particles has both an onshore-offshore and a longshore com- 
ponent. Onshore-offshore transport is usually the most significant type 
of transport in the offshore zone, except in regions of strong tidal 
currents. Both longshore and onshore-offshore transport are significant 
in the surf zone. 

Engineering problems involving littoral transport generally require 
answers to one or more of the following questions: 

(a) What are the longshore transport conditions at the site? 
(Needed for the design of groins, jetties, navigation channels, and 
inlets.) 

Cb) What is the trend of shoreline migration over short and long 
time intervals? (Needed for design of coastal structures, including 
navigation channels.) 

(c) How far seaward is sand actively moving? (Needed in the design 
of sewage outfalls and water intakes.) 

(d) What is the direction and rate of onshore-offshore sediment 
motion? (Needed for sediment budget studies and beach fill design.) 

(e) What is the average shape, and the expected range of shapes, 
for a given beach profile? (Needed for design of groins, beach fills, 
navigation structures and flood protection.) 

(f) What effect will a postulated structure or project have on 
adjacent beaches and on littoral transport? (Needed for design of all 
coastal works.) 

This section presents recommended methods for answering these and 
related questions. The section indicates accepted practice based on 
field observations and research results. Section 4.52 deals with onshore- 
offshore transport, presenting material pertinent to answering questions 
(b) through (f). Section 4.53 deals with longshore transport, presenting 
material pertinent to questions (a) , (b) , and (f) . 



4-52 



4.512 Zones of Transport . Littoral transport occurs in two modes: hed- 
toad transport J the motion of grains rolled over the bottom by the shear 
of water moving above the sediment bed; and suspended-load transport j the 
transport of grains by currents after the grains have been lifted from 
the bed by turbulence. 

Both modes of transport are usually present at the same time, but it 
is hard to distinguish where bedload transport ends and suspended-load 
transport begins. It is more useful to identify two zones of transport 
based on the type of fluid motion initiating sediment motion: the off- 
shore zone where transport is initiated by wave-induced motion over rip- 
ples, and the surf zone where transport is initiated by the passing break- 
er. In either zone, net sediment transport is the product of two pro- 
cesses: the periodic wave-induced fluid motion that initiates sediment 
motion, and the superimposed currents (usually weak) which transport the 
sediment set in motion. 

a. Offshore Zone . Waves traveling toward shallow water eventually 
reach a depth where the water motion near the bottom begins to affect the 
sediment on the bottom. At first, only low-density material (such as sea- 
weed and other organic matter) moves. This material oscillates back and 
forth with the waves, often in ripple-like ridges parallel to the wave 
crests. For a given wave condition, as the depth decreases, water motion 
immediately above the sediment bed increases until it exerts enough shear 
to move sand particles. The sand then forms ripples with crests parallel 
to the wave crests. These ripples are typically uniform and periodic, 
and sand moves from one side of the crest to the other with the passage 
of each wave. 

As depth decreases to a value several times the wave height, the veloc- 
ity distribution with time changes from approximately sinusoidal to a dis- 
tribution that has a high shoreward component associated with the brief 
passage of the wave crest, and lower seaward velocities associated with 
the longer time interval occupied by the passage of the trough. As the 
shoreward water velocity associated with the passing crest decreases and 
begins to reverse direction over a ripple, a cloud of sand erupts upward 
from the lee (landward) side of the ripple crest. This cloud of sand 
drifts seaward with the seaward flow under the trough. At these shallow 
depths, the distance traveled by the cloud of suspended sediment is two 
or more ripple wavelengths, so that the sand concentration at a point 
above the ripples usually exhibits at least two maximums during the pass- 
age of the wave trough. These maximums are the suspension clouds shed by 
the two nearest upstream ripples. The approach of the next wave crest 
reverses the direction of the sand remaining suspended in the cloud. The 
landward flow also drags material shoreward as bedload. 

For the nearshore profile to be in equilibrium with no net erosion or 
accretion, the average rate at which sand is carried away from a point on 
the bottom must be balanced by the average rate at which sand is added. 
Any net change will be determined by the net residual currents near the 
bottom which transport sediment set in motion by the waves. These currents, 

4-53 



the subject of Section 4.4, include longshore currents and mass-transport 
currents in the on shore -offshore direction. It is possible to have ripple 
forms moving shoreward while residual currents above the ripples carry 
suspended sediment clouds in a net offshore direction. Information on the 
transport of sediment above ripples is given in Bijker (1970), Kennedy and 
Locher (1972), and Mogridge and Kamphuis (1972). 

b. Surf Zone. The stress of the water on the bottom due to turbu- 
lence and wave- induced velocity gradients moves sediment in the surf zone 
with each passing breaker crest. This sediment motion is both bedload 
and suspended- load transport. Sediment in motion oscillates back and 
forth with each passing wave, and moves alongshore with the longshore 
current. On the beach face, the landward termination of the surf zone, 
the broken wave advances up the slope as a bore of gradually decreasing 
height, and then drains seaward in a gradually thinning sheet of water. 
Frequently, the draining return flows in gullies and carries sediment to 
the base of the beach face. 

In the surf zone, ripples cause significant sediment suspension, but 
here there are additional eddies caused by the breaking wave. These eddies 
have more energy and are larger than the ripple eddies. The greater energy 
suspends more sand in the surf zone than offshore. The greater eddy size 
mixes the suspended sand over a larger vertical distance. Since the size 
is about equal to the local depth, significant quantities of sand are sus- 
pended over most of the depth in the surf zone. 

Since breaking waves suspend the sediment, the amount suspended is 
partly determined by breaker type. Data from Fairchild (1972, Figure 5), 
show that spilling breakers usually produce noticeably lower suspended 
sediment concentrations than do plunging breakers. See Fairchild (1972) 
and Watts (1953) for field data; Fairchild (1956 and 1959) for lab data. 
Typical suspended concentrations of fine sand range between 20 parts per 
million and 2 parts per thousand by weight in the surf zone, and are about 
the same near the ripple crests in the offshore zone. 

Studies of suspended sediment concentrations in the surf zone by 
Watts (1953) and Fairchild (1972) indicate that sediment in suspension in 
the surf zone may form a significant portion of the material in longshore 
transport. However, present understanding of sediment suspension, and 
the practical difficulty of obtaining and processing sufficient suspended 
sediment samples have limited this approach to predicting longshore trans- 
port. 

4.513 Profiles . Profiles are two-dimensional vertical sections showing 
how elevation varies with distance. Coastal profiles (See Figs. 4-1 and 
4-18) are usually measured perpendicular to the shoreline, and may be 
shelf profiles, nearshore profiles, or beach profiles. Changes on near- 
shore and beach profiles are interrelated, and are highly important in 
the interpretation of littoral processes. The measurement and analysis 
of combined beach and nearshore profiles is a major part of most engineer- 
ing studies of littoral processes. 



4-54 



30 

20 

10 





V 






beacn Krotiie 


1 


\ 




(horiz 


ontol scale divided by l( 
jrtical exageration = 2) 


DO) 


V 




(V( 




N 

































~ ^^^ 


MSL 












" ^ 






E 

3 



P 20 



< 



o 

c 
o 







20 









Nearshore 


Profile 








(hori 


zontal scale divided by 
rtical exageration = 10) 


0) 


V 




(ve 




X 


MSL 










^v^^ 














^ ^ 


■- 












^ 


■ . 


-— ..^___ 



o 
u 



> 






MSL 




inner 


Continental Shelf Pro 
rtical exageration =50! 


hie 


\ 




(ve 




-40 


























-80 



























-120 































10,000 20,000 

Horizontal Distance (Arbitrary Origin), feet 



30,000 



Figure 4-18. Three Scales of Profiles, Westhampton, Long Island 



4-55 



a. Shelf Profiles . The shelf profile is typically a smooth, concave- 
up curve showing depth to increase seaward at a rate that decreases with 
distance from shore. (bottom profile in Figure 4-18.) The smoothness of 
the profile may be interrupted by other superposed geomorphic features, 
such as linear shoals. (Duane, et al., 1972.) Data for shelf profiles 
are usually obtained from charts of the National Ocean Survey (formerly, 
U.S. Coast and Geodetic Survey). 

The measurable influence of the shelf profile on littoral processes 
is largely its effect on waves. To an unknown degree, the shelf may also 
serve as a source or sink for beach sand. Geologic studies show that 
much of the outer edge of a typical shelf profile is underlain by rela- 
tively coarse sediment, indicating a winnowing of fine sizes. (Dietz, 
1963; Milliman, 1972; and Duane, et al . , 1972.) Landward from this resi- 
dual sediment, sediment often becomes finer before grading into the rela- 
tively coarser beach sands. 

b. Nearshore Profiles . The nearshore profile extends seaward from 
the beach to depths of about 30 feet. Prominent features of most near- 
shore profiles are longshore bars; see middle profile of Figure 4-18 and 
Section 4.525. In combination with beach profiles, repetitive nearshore 
profiles are used in coastal engineering to estimate erosion and accre- 
tion along the shore, particularly the behavior of beach fill, groins, 
and other coastal engineering structures. Data from nearshore profiles 
must be used cautiously. (see Section 4.514.) Under favorable condi- 
tions nearshore profiles have been used in measuring longshore transport 
rates. (Caldwell, 1956.) 

c. Beach Profiles . Beach profiles extend from the foredunes, cliffs, 
or mainland out to mean low water. Terminology applicable to features of 
the beach profile is in Appendix A (especially Figures A-1 and A-2). The 
backshore extends seaward to the foreshore, and consists of one or more 
berms at elevations above the reach of all but storm waves. Berm sur- 
faces are nearly flat and often slope landward at a slight downward angle. 
(See Figure 4-1.) Berms are often bounded on the seaward side by a break 
in slope known as the berm crest. 

The foreshore is that part of the beach extending from the highest 
elevation reached by waves at normal high tide seaward to the ordinary 
low water line. The foreshore is usually the steepest part of the beach 
profile. The boundary between the backshore and the foreshore may be 
the crest of the most seaward berm, if a berm is well developed. The 
seaward edge of the foreshore is often marked by an abrupt step at low 
tide level. 

Seaward from the foreshore, there is usually a low-tide terrace which 
is a nearly horizontal surface at about mean low tide level. (Shepard, 
1950; and Hayes, 1971.) The low-tide terrace is commonly covered with sand 
ripples and other minor bed forms, and may contain a large bar-and-trough 
system, which is a landward-migrating sandbar (generally parallel to the 
shore) common in the nearshore following storms. Seaward from the low-tide 



4-56 



terrace (seaward from the foreshore, if the low-tide terrace is absent) 
are the longshore troughs and longshore bars. 

4.514 Profile Accuracy . Beach and nearshore profiles are the major 
source of data for engineering studies of beach changes; sometimes lit- 
toral transport can be estimated from these profiles. Usually, beach 
and nearshore profiles are measured at about the same time, but differ- 
ent tecliniques are needed for their measurement. The nearshore profile 
is usually measured from a boat or amphibious craft, using an echo sounder 
or leadline, or from a sea sled. (Kolessar and Reynolds, 1965-66; and 
Reimnitz and Ross, 1971.) Beach profiles are usually surveyed by standard 
leveling and taping techniques. 

The accuracy of profile data is affected by four types of error: 
sounding error, spacing error, closure error, and error due to temporal 
fluctuations in the sea bottom. These errors are more significant for 
nearshore profiles than for beach profiles. 

Saville and Caldwell (1953) discuss sounding and spacing errors. 
Sounding error is the difference between the measured depth and the 
actual depth. Under ideal conditions, average sonic sounding error may 
be as little as 0.1 foot, and average leadline sounding error may be 
about twice the sonic sounding error. (Saville and Caldwell, 1953.) (This 
suggests that sonic sounding error may actually be less than elevation 
changes caused by transient features like ripples. Experience with suc- 
cessive soundings in the nearshore zone indicates that errors in practice 
may approach 0.5 foot.) Sounding errors are usually random and tend to 
average out when used in volume computations, unless a systematic error 
due to the echo sounder or tide correction is involved. Long-period 
water level fluctuations affect sounding accuracy by changing the water 
level during the survey. At Santa Cruz, California, the accuracy of 
hydrographic surveys was ±1.5 feet due to this effect. (Magoon, 1970.) 

Spacing error is the difference between the actual volume of a seg- 
ment of shore and the volume estimated from a single profile across that 
segment. Spacing error is potentially more important than sounding error, 
since survey costs of long reaches usually dictate spacings between near- 
shore profiles of thousands of feet. For example, if a 2-mile segment of 
shore 4,000 feet wide is surveyed by profiles on 1,000-foot spacings, then 
the spacing error is about 9 cubic yards per foot of beach front per survey, 
according to the data of Saville and Caldwell (1953, Figure 5). This error 
equals a major part of the littoral budget in many localities. 

Closure error arises from the assumption that the outer ends of 
nearshore profiles have experienced no change in elevation between two 
successive surveys. Such an assumption is often made in practice, and 
may result in significant error. An uncompensated closure error of 0.1 
foot, spread over 1,000 feet at the seaward end of a profile, implies a 
change of 3.7 cubic yards per time interval per foot of beach front where 



4-57 



the time interval is the time between successive surveys. Such a volume 
change may be an important quantity in the sediment budget of the litto- 
ral zone. 

A fourth source of error comes from assuming that the measured beach 
profiles (which are only an instantaneous picture), represent a long-term 
condition. Actually, beach and nearshore profiles change rapidly in re- 
sponse to changing wave conditions, so that differences between succes- 
sive surveys of a profile may merely reflect temporary differences in 
bottom elevation caused by storms and seasonal changes in wave climate. 
Such fluctuations obliterate long-term trends during the relatively short 
time available to most engineering studies. This fact is illustrated for 
nearshore profiles by the work of Taney (1961a, Appendix B) who identified 
and tabulated 128 profile lines on the south shore of Long Island that had 
been surveyed more than once from 192 7 to 1956. Of these, 47 are on 
straight shorelines away from apparent influence by inlets, and extend 
from Mean Low Water (MLW) to about -30 feet MLW. Most of these 47 pro- 
files were surveyed three or more times, so that 86 separate volume 
changes are available. These data lead to the following conclusions: 

(a) The net volume change appears to be independent of the time 
between surveys, even though the interval ranged from 2 months to 16 
years. (See Figure 4-19.) 

(b) Gross volume changes (the absolute sums of the 86 volume changes) 
are far greater than net voliome changes (the algebraic sums of the 86 vol- 
lome changes). The gross volume change for all 86 measured changes is 8,113 
cubic yards per foot; the net change is -559 cubic yards per foot (loss in 
volume) . 

(c) The mean net change between surveys, averaged over all pairs of 
surveys, is -559/86 or -6.5 cubic yards per foot of beach. The median 
time between surveys is 7 years, giving a nominal rate of volume change 
of about -1 cubic yard per year per foot. 

These results point out that temporary changes in successive surveys 
of nearshore profiles are usually much larger than net changes, even when 
the interval between surveys is several years. These data show that care 
is needed in measuring nearshore profiles if results are to be used in 
engineering studies. The data also suggest the need for caution in inter- 
preting differences obtained in two surveys of the same profiles. 

The positions of beach profiles must be marked so that they can be 
recovered during the life of the project. The profile monuments should 
be tied in by survey to local permanent references. If there is a long- 
term use for data at the profile positions, the monuments should be ref- 
erenced by survey to a state coordinate system or other reference system, 
so that the exact position of the profile may be recovered in the future. 
Even if there is no anticipated long-term need, future studies in any 
coastal region are likely, and will benefit greatly from accurately sur- 
veyed, retrievable benchmarks. 



4-58 



500- 



>> 

to 

>» 

V 

> 
k. 

3 

c 
« 



.c 

u 

« 

E 

_3 
O 
> 



(jt>- 



oo o 



-^ 



I I I I l l 






I I I I I I I 



100 



J. 

8 



200 



c 
13 



■500- 



Time Between Surveys, months 

( based on doto from Toney, 1961 a ) 



Figure 4-19. Unit Volume Change Versus Time Between Surveys for Profiles 
on South Shore of Long Island. Data from Profiles Extending 
from MSL to about the -30 depth Contour. 



4-59 



For coastal engineering, the accuracy of shelf profiles is usually 
less critical than the accuracy of beach and nearshore profiles. Gener- 
ally, observed depth changes between successive surveys of the shelf do 
not exceed the error inherent in the measurement. However, soundings 
separated by decades suggest that the linear shoals superposed on the 
profile do show small but real shifts in position. (Moody, 1964, p. 143.) 
Charts giving depths on the continental shelves may include soundings 
that differ by decades in date. 

Plotted profiles usually use vertical exaggeration or distorted 
scales to bring out characteristic features. This exaggeration may lead 
to a false impression of the actual slopes. As plotted, the three pro- 
files in Figure 4-18 have roughly the same shape, but this sameness has 
been obtained by vertical exaggerations of 2x, lOx, and 50x. 

Sand level changes in the beach and nearshore zone may be measured 
quite accurately from pipes imbedded in the sand. (Inman and Rusnak, 1956; 
Urban and Galvin, 1969; and Gonzales, 1970.) 

4.52 ONSHORE -OFF SHORE TRANSPORT 

4,521 Sediment Effects . Properties of individual particles which have 
been considered important in littoral transport include: size, shape, 
immersed specific gravity, and durability. Collections of particles 
have the additional properties of size distribution, permeability, and 
porosity. These properties determine the forces necessary to initiate 
and maintain sediment motion. 

For typical beach sediment, size is the only property that varies 
greatly. However, quantitative evaluation of the size effect is usually 
lacking. A gross indication of a size effect is the accumulation of 
coarse sediment in zones of maximum wave energy dissipation, and deposi- 
tion of fine sediment in areas sheltered from wave action. (e.g. King, 
1972, pp. 302, 307, 426.) Sorting by size is common over ripples (Inman, 
1957) and large longshore bars (Saylor and Hands, 1970). Field work on 
size effects in littoral transport does not permit definite conclusions. 
(King, 1972, p. 483; Inman, Komar, and Bowen, 1969; Castanho, 1970; Ingle, 
1966, Figure 112; Yasso, 1962; and Zenkovich, 1967a.) 

The shape of most littoral materials is approximately spherical; 
departures from spherical are usually too slight to affect littoral 
transport. 

Immersed specific gravity (specific gravity of sediment minus spec- 
ific gravity of fluid) is theoretically an important physical property of 
the sediment particle. (Bagnold, 1963.) However, the variation in immersed 
specific gravity for typical littoral materials in water is small since 
most beach sediments are quartz (immersed specific gravity = 1.65), and 
most of the remainder are calcium carbonate (immersed specific gravity 
= 1.9). Thus, little variation in littoral transport is expected from 
variation in immersed specific gravity. 

4-60 



Durability (resistance to abrasion, crushing, and solution) is usu- 
ally not a factor within the lifetime of an engineering project. (Kuenen, 
1956; Rusnak, Stockman, and Hofmann, 1966; and Thiel, 1940.) Possible 
exceptions may include basaltic sands on Hawaiian beaches (Moberly, 1968), 
some fragile carbonate sands which may be crushed to finer sizes when sub- 
ject to traffic, (Duane and Meisburger, 1969, p. 44), and carbonate sands 
which may be soluble imder some conditions. (Bricker, 1971.) In general, 
recent information lends further support to the conclusion of Mason (1942) 
that, "On sandy beaches the loss of material ascribable to abrasion . . . 
occurs at rates so low as to be of no practical importance in shore pro- 
tection problems." 

Size distribution and its relation to sediment sorting may be impor- 
tant for design of beach fills. (See Sections 5.3 and 6.3.) Permeability 
and porosity affect energy dissipation (Bretschneider and Reid, 1954; Bret- 
schneider, 1954) and wave runup. (See Section 7.21; and Savage, 1958.) 

Sediment properties are physically most important in determining fall 
velocity and the hydraulic roughness of the sediment boundary. Fall velo- 
city effects are important in onshore-offshore transport. Hydraulic rough- 
ness effects have been insufficiently studied, but they appear to affect 
initiation of sediment transport and energy dissipation. This may be par- 
ticularly true for flow of the swash on the plane surface of the foreshore. 
(Everts, 1972.) 

4.522 Initiation of Sediment Motion . Considerable hydraulic and coastal 
engineering research has been devoted to the initiation of sediment motion 
under moving water. From this research has come general agreement (Graf, 
1971, Chapter 6; Hjulstrom, 1939; and Everts, 1972) that the initiation of 
motion on a level bed of fine or medium sand requires less shear (lower 
velocities) than the initiation of motion on a level bed of silt or gravel; 
(Figure 4-7 for size classes). It is also generally agreed that critical 
entraining velocities for sand are usually less than 1 foot per second. 

Velocities of wave-induced water motion in the offshore zone can be 
estimated fairly well from the equation of small -amplitude theory. (See 
Chapter 2.) This theory leads to Equation 2-13 which can be transformed 
into a dimensionless expression for velocity at the sand surface (z = -d) 

max ( j-> TT , ^ 

y '^> , (4-19) 



H 



sinh 2^<y^ 



which is plotted in dimensionless form in Figure 4-20 and for common values 
of wave period in Figure 4-21. Figures 4-20 and 4-21 give maximum bottom 
particle velocity, ^imax{-d) > ^^ ^ function of depth, wave period, and 
local wave height. This prediction by linear theory for the offshore zone, 
and the related results from solitary-wave theory for the zone near break- 
ing, agree fairly well with measurement in the field. (Inman and Nasu, 
1956; and Cook and Gorsline, 1972, Figures 5 and 6.). 



4-61 



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4-62 



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4-63 



************** EXAMPLE PROBLEM 



*************** 



GIVEN: A wave in depth d = 200 feet, with period T = 9 seconds, and a 
maximum bottom velocity umaxr-d) 2: 1.0 foot per second. 

FIND : The minimxim wave height that will create the given bottom velocity. 

SOLUTION : Calculate 



Lo = 



and 



lit 

= 5.12(9)2 
= 415 feet . 

200 



Lo 415 



= 0.482 



Entering Figure 4-20 with d/Lo = 0.482, determine 



u T 
H 

H 



= 0.30 , 



"ma.C(_jjT 



0.30 



H =(l)(2) = 3ofeee. 
0.30 

Thus a 30-foot minimum wave height with a 9-second wave period is needed 
to create a bottom velocity equal to or greater than 1 foot per second 
in 200 feet of water. Alternatively, a curve for a 9-second period can 
be interpolated in Figure 4-21 and uma«(-tf)T/H can be read from the 
curve's intersection with the 200-foot depth. 

************************************* 

As a wave moves into shallower water, both bottom velocity and size 
of water-particle orbit increase. For some low velocities where the hori- 
zontal orbit is small, sand moves as individual grains rolling across the 
surface, but for most conditions, once quartz sand begins to move, ripples 
form (Kennedy and Falcon, 1965; Carstens, et al., 1969; Cook, 1970). Thus, 
the initiation of sediment motion is usually indicated by the formation 
of sediment ripples. 



4-64 



Figure 4-22, from Inman (1957) and including data from two earlier 
studies, shows the velocity needed to start motion in a sediment bed of 
a given grain size. These results are in general agreement with other 
studies relating critical velocity to grain size. Also shown in this 
figure are maximum velocities above which ripples tend to be smoothed 
off, in qualitative agreement with conditions for bed forms in unidirec- 
tional flows. (Southard, 1972.) 

From Figure 4-22, it appears that maximum wave-induced bottom veloc- 
ities between 0.4 and 1.0 foot per second are sufficient to initiate sand 
motion imder waves. In field studies, Inman (1957) found that ripples are 
always present whenever computed maximum velocities exceed 0.33 foot per 
second, and Cook and Gorsline (1972) report ripples above a velocity range 
of 0.5 to 0.6 foot per second. Equation 4-19 can be used to determine the 
combination of wave conditions and depth that produces any given critical 
value of Umax at the bottom. Figure 4-23 shows the relation between depth 
and wave height, for given wave periods, for a critical velocity, vmax = 
0.5 foot per second. 

4.523 Seaward Limit of Significant Transport . Figures 4-20 through 4-23 
and a knowledge of offshore wave climate suggest that waves can move bot- 
tom sediments over most of the Continental Shelf (to depths of 100 to 400 
feet or more) during some time of the year. Geologic studies indicate 
that fine material has been winnowed from the surficial sediments over 
much of the shelf. (Shepard, 1963; and Dietz, 1963.) The question is, 
what is the maximum depth to which the rate of sand movement is signifi- 
cant in coastal engineering? This section discusses field data that 
supply partial answers to this question. 

a. Bathymetry . Dietz (1963) and others point out that waves rework 
nearshore sands, smoothing out irregularities by longshore and onshore- 
offshore transport. This smoothing produces a quasi -equilibrium surface 
in the nearshore zone which forms relatively straight contours, nearly 
parallel to the shoreline. 

Most bathymetric charts with closely spaced contours as illustrated 
by Figures 4-24 and 4-25 show that isobaths near the shore run parallel 
to the shoreline; further offshore, the contours may indicate linear 
shoals (Duane, et al., 1972), or other irregular submarine features. 

Following the idea of Dietz (1963), the depth below which shore- 
parallel contours give way to irregular contours is assumed to mark the 
local transition between the nearshore zone where sands are moved by the 
waves in significant quantities and the offshore zone where sand is moved 
in lesser quantities. Possible exceptions to this shore-parallel contour 
rule are the contours around river deltas and inlets, or where reefs and 
ledges occur in the nearshore zone. 



4-65 




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4-66 



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of 0.5 Ft/Sec ( Based on Linear Theory ) 



4-67 



e5'>4e 



30'I0 




Contour Intervol 5 feet 
up to 40, 2 feet thereofter 



Feet 

2000 4000 



(from Dietz, 1963, p. 984) 



Figure 4-24. 



Nearshore Bathymetry with Shore-Parallel 
Contours off Panama City, Florida 



Bathymetry, such as that in Figures 4-24 and 4-25, suggests that the 
depth to the deepest shore-parallel contour is usually constant along the 
shore for distances of several miles, but that this depth may vary with 
longshore distances of about 10 miles. (See Figure 4-25.) The depth to 
the deepest shore-parallel contour may depend on the contour spacing, but 
this is not important if contour intervals are small relative to the total 
depths involved. In general, the deepest shore-parallel contour is between 
15 and 60 feet. In most localities, this depth is somewhat deeper than 
that at which nearshore profiles are presumed to close-out. These near- 
shore contours probably reflect longer term adjustment to extreme storms 
that occur rarely during the typical time interval between repetitive 
nearshore profile surveys. 

b. Size Distribution . Geologic studies (Milliman, 1972; and Curray, 
1965) suggest that littoral sands grade seaward into finer materials before 
the relatively coarse sands of the shelf are reached. In some places the 
boundary between the coarser shelf sediment and this finer material is quite 
sharp. (Pilkey and Frankenberg, 1964.) The finer material is currently 



4-68 




4-69 



interpreted as bounding the seaward edge of sediment moved by waves in sig- 
nificant quantities. This band of finer material suggests that there is 
little exchange between littoral and shelf sands in most places. 

c. Sand Budget Balancing . Onshore sand transport has been suggested 
as a source of sand for several coastal localities that lack other obvious 
sand sources. (Shepard, 1963, pp. 176-177; and Pierce, 1969.) For example. 
Pierce suggests that the offshore must supply 440,000 cubic yards per year 
of sand to the southern segment of the Outer Banks of North Carolina be- 
cause other known sources do not balance the budget. 

d. Transport in Nearshore Zone . Although theoretical, experimental, 
and field data show that waves move sand some of the time over most of the 
Continental Shelf, most of the data suggest that sand from the shelf is not 
a significant contributor to the sediment budget of the littoral zone. 

Sand transport from the nearshore zone is more likely. Surveys show 
that sand in the nearshore zone does move, although it is difficult to meas- 
ure direction of motion. The presence of shore-parallel contours along most 
open shores is evidence that the waves actively mold the sand bottom in the 
nearshore zone. However, the time scale of transport in the nearshore zone 
may be relatively slow. 

In tests at Santa Barbara, California, and at Atlantic City and Long 
Branch, New Jersey, dredged sands were dumped offshore in depths ranging 
from 15 to 40 feet, but no measurable onshore migration of the sand re- 
sulted for times of about 1 year. (Hall and Herron, 1950.) Radioactive 
tracers have shown that gravel moves slowly landward in 30 feet of water 
at a rate of about 0.5 cubic yard per year per foot of beach. (Crickmore 
and Waters, 1972.) 

At shallower depths in the nearshore zone, onshore sand transport 
following storms is well documented. Transport of sediment suspended 
over ripples by the mass transport velocity is more than adequate to 
return sand eroded from the beach or to transport sand eroded from the 
nearshore bottom to the beach. 

e. Summary on Seaward Limit . The deepest shore-parallel contour 
appears to be a usable estimate of the maximum depth where significant 
sand transport can be expected. This depth varies from 15 feet to per- 
haps 60 feet or more along U.S. coasts. This choice may be modified for 
specific wave conditions using Figure 4-23 to find the depth where maxi- 
mum wave-induced velocity first exceeds 0.5 foot per second. If this 
depth is less than the depth of the deepest shore-parallel contour, it 
should be used as the seaward limit for the given wave conditions, 

4.524 Beach Erosion and Recovery . 

a. Beach Erosion . Beach profiles change frequently in response to 
winds, waves, and tides. Profiles are also affected by events in the long- 
shore direction that influence the longshore transport of sand. The most 



4-70 



notable rapid rearrangement of a profile is by storm waves, especially 
during storm surge (Section 3.8) which enables the waves to attack at 
higher elevations on the beach, (see Figure 1-7.) 

The part of the beach washed by runup and runback is the beach face. 
Under normal conditions, the beach face is contained within the fore- 
shore, but during storms, the beach face is moved shoreward by the cut- 
ting action of the waves on the profile. The waves during storms are 
steeper, and the runback of each wave on the beach face carries away more 
sand than is brought to the beach by the runup of the next wave. Thus 
the beach face migrates landward, cutting a scarp into the berm. (See 
Figure 1-7.) 

In moderate storms, the storm surge and accompanying steep waves will 
subside before the berm has been significantly eroded. In severe storms, or 
after a series of moderate storms, the backshore may be completely eroded, 
after which the waves will begin to erode the coastal dunes, cliffs, or main- 
land behind the beach. 

The extent of storm erosion depends on wave conditions, storm surge, 
the stage of the tide and storm duration. 

Potential damage to property behind the beach depends on all these 
factors and on the volume of sand stored in the beach-dune system when a 
storm occurs. 

For planning and design purposes, it is useful to know the magnitude 
of beach erosion to be expected during severe storms. Table 4-5 tabulates 
the effect of four notable extratropical storms along the Atlantic coast 
of the U.S. This table provides information on typical observed order-of- 
magnitude values for beach erosion above mean sea level (MSL) from single 
storms. 

For the storm of 17 December 1970, information is available from 
seven localities (Column 2 of Table 4-5). (DeWall, et al. , 1971.) The 
three other storms include two closely spaced storms affecting Jones Beach, 
New York, in February 1972 (Everts, 1972), and a storm that affected the 
northern New Jersey coast in November 1953. (Caldwell, 1959.) Character- 
istics that distinguish one storm from another are duration and storm 
surge. (See Columns 9 and 10, Table 4-5.) Storm waves lasted about 1 day 
for the 17 December 1970 storm, and about 2 days for the other three 
storms. (See Column 9.) Storm surge elevation varied from a low of 2.8 
feet to a high of 6 feet in the New York Bight area. The November 1953 
storm combined longer duration and high storm surge; the 17 December 1970 
storm had short duration and moderate storm surge; and the February 1972 
storms both had longer duration, one with moderate storm surge and the 
other with low storm surge. 

Duration and storm surge (Columns 9 and 10) are consistent with storm 
damage data (Columns 11 through 16, although the effect is influenced by 
the choice of profiles included in each study. The December 1970 storm 



4-71 







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4-73 



includes all profiles surveyed. The two February 1972 storms at Jones 
Beach include all profiles away from the influence of inlets, reducing 
the number of profiles from 15 in the December 1970 storm to 10 in the 
February 1972 storms. The November 1953 storm gives relatively high 
storm damage which may partly result from the long time interval between 
pre-storm survey and the storm. (See Columns 1 and 3.) These results 
are also affected by the fact that they omit some profiles "believed to 
be influenced by the presence of a seawall or a bulkhead," (Caldwell, 
1959, p. 4.) 

Although the data in Table 4-5 are not completely comparable, the 
results do suggest that the average volume of sand eroded above mean sea 
level from beaches about 5 or more miles long has a certain range of val- 
ues. A moderate storm may remove 4 to 10 cubic yards per foot of beach 
front above MSL; an extreme storm (or a moderate storm that persists for 
a long time) may remove 10 to 20 cubic yards per foot; rare storms that 
are most destructive in beach erosion due to a combination of intensity, 
duration, and orientation may remove 20 to 50 cubic yards per foot. These 
values are average for beaches 5 to 10 miles or more long, and they are 
compatible with other, less complete, data for notable storms. (Caldwell, 
1959; Shuyskiy, 1970; and Harrison and Wagner, 1964.) For comparative 
purposes, a berm 100 feet wide at an elevation of 10 feet MSL contains 37 
cubic yards per foot of beach front, a quantity that would be adequate 
except for extreme storms. 

In terms of horizontal changes rather than the volume changes in 
Table 4-5, a moderate storm can erode a typical beach 75 to 100 feet or 
more, and leave it exposed to greater erosion if a second storm follows 
before the beach has recovered. This possibility should be considered 
in design and placement of beach fills and other protective measures. 

Extreme values of erosion may be more useful than mean values for 
design. Column 17 of Table 4-5 suggests that the ratio of the most ero- 
ded above-MSL-profile to the average profile for east coast beaches ranges 
from about 1.5 to 6. If the average erosion per profile is based only on 
those profiles showing net erosion, then this ratio is probably between 
1.5 and 3. 

Although the dominant result of storms on the above MSL part of 
beaches is erosion, most post-storm surveys show that the storm produces 
local accretion as well. Of the 90 profiles from Cape Cod, Massachusetts, 
to Cape May, New Jersey, surveyed immediately after the December 1970 
storm, 16 showed net accretion above mean sea level. (Compare Columns 4, 
11, and 12 in Table 4-5.) Similar results are indicated for a number of 
more severe storms. (Caldwell, 1959.) 

The storm surveys also show that the shoreline on many beaches may 
prograde seaward even though the profile as a whole loses volume, or 
vice versa. This possibility suggests caution in interpreting aerial 
photos of storm damage. (Everts, 1973.) 



4-74 



b. Beach Recovery . The typical beach profile left by a severe storm 
is a simple, con cave -upward curve extending seaward to low tide level or 
below, (See top of Figure 4-26.) The sand that has been eroded from the 
beach is deposited mostly as a ramp or bar in the surf zone that exists at 
the time of the storm. Immediately after the storm, beach repair begins 
by a process that has been documented in detail, (e.g., Hayes, 1971; Davis, 
et al., 1972; Davis and Fox, 1972; and Sonu and van Beek, 1971.) Sand that 
has been deposited seaward of the shoreline during the storm begins moving 
landward as a sandbar with a gently sloping seaward face and a steeper land- 
ward face. (See Figure 4-26.) These bars have associated lows (riinnels) 
on the landward side and occasional drainage gullies across them. (King, 
1972, p. 339.) These systems are characteristic of post-storm beach accre- 
tion under a wide range of wave, tide, and sediment conditions. (Davis, 
et al., 1972.) They are sometimes called ridge -and- runnel systems. 

The processes of accretion occur as follows. Sand is transported 
landward over the nearly flat seaward face of the bar by the waves. At 
the bar crest, the sand avalanches down the landward slip face. If the 
process continues long enough, the bar reaches the landward limit of 
storm erosion where it is "welded" onto the beach. (e.g. Davis, et al., 
1972.) Further accretion continues by adding layers of sand to the top 
of the bar which, by then, is a part of the beach. (See Figure 4-27.) 

Berms may form immediately on a post-storm profile without an inter- 
vening bar-and-trough, but the mode of berm accretion is quite similar 
to the mode of bar-and-trough growth. Accretion occurs both by addition 
of sand laminae to the beach face (analogous to accretion on the seaward- 
dipping top of the bar in the bar-and-trough) and by addition of sand on 
the slight landward slope of the berm surface when waves carrying sedi- 
ment overtop the berm crest (analogous to accretion on the landward dip- 
ping slip face of the bar). This process of berm accretion is also illus- 
trated in Figure 4-1. 

The rate at which the berm builds up or the bar migrates landward to 
weld onto the beach varies greatly, apparently in response to: wave con- 
ditions, beach slope, grain size, and the length of time the waves work 
on the bars. (Hayes, 1971.) Compare the slow rate of accretion at Crane 
Beach in Figure 4-26 (mean tidal range 9 feet, spring range 13 feet) with 
the rapid accretion on the Lake Michigan shore in Figure 4-2 7 (tidal range 
less than 0.25 foot). 

Post-storm studies by CERC show that the rate of post-storm replenish- 
ment by bar migration and berm building is usually rapid immediately after 
the storm. T^his rapid buildup is important in evaluating the effect of 
severe storms because (unless surveys are made within hours after the 
storm) the true extent of erosion during the storm is likely to be ob- 
scured by the post-storm recovery. Lack of surveys before the start of 

post-storm recovery may explain some survey data that show MSL accretion 
on profiles that have lost volume. 

4-75 



^» 



\ 
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10 n 

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Summer Accretion 29 May -7 September 1967 

Station CBA, Crane Beach 

Ipswich, Massachusetts 






'• — ' 



-29 Moy 

-I 3 June 
^ __M_LW_ 
— I July 



,\ I July 

-20 July 

-24 July 

-"3 Aug 
_8 Aug 



14 Aug 



22 Aug 



30 Aug 



7Sept„LW 



100 



200 300 

Feet 



400 500 

(from Hayes, 1972) 



Figure 4-26. Slow Accretion of Ridge- and- Runnel at Crane 
Beach, Massachusetts 



4-76 




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4-77 



The ideal result of post-storm beach recovery is a wide backshore 
that will protect the shore from the next storm. Beach recovery may be 
prevented when the period between successive storms is too short. Main- 
tenance of coastal protection requires knowledge of the necessary width 
and elevation of the backshore appropriate to local conditions, and 
adequate surveillance to determine when this natural sand reservoir is 
diminished to a point where it may not protect the backshore during the 
next storm. 

4.525 Bar-Berm Prediction . High, steep waves scour the beach, eroding the 
foreshore into a simple con cave -upwards profile. The material eroded from 
the beach is deposited offshore as a longshore bar. Waves of low steepness 
tend to push sand onto the beach, usually as migrating longshore-bar sys- 
tems which eventually become part of the beach. In contrast to the concave- 
up eroded profile discussed previously, the accreted profile is concave- 
downward. Idealized eroded and accreted profiles (measured in a prototype- 
scale wave tank) across the beach and nearshore zone are shown in Figure 
4-28. 

To design a beach that contains a reservoir of sand in the backshore 
sufficient to survive a design storm, a minimum requirement is the ability 
to distinguish wave conditions that cause eroded profiles from those that 
cause accretion. Usually, it is assiomed that a berm characterizes an ac- : 
creted profile and that a bar characterizes an eroded profile. (See Figure 
4-28.) Tliis picture is somewhat idealized. A sharp berm crest between 
backshore and foreshore is often lacking, and on some beaches the berm is 
absent, so that the top of the foreshore reaches the dune or cliff line. 
Berms are illustrated in Figures 4-1 and 4-27. 

Similarly, the idealized longshore bar seaward of an eroded beach, 
(middle profile of Figure 4-18) is often absent, and in its place there 
may be several subdued bars or a platform extending to the breaker line 
at nearly constant depth. 

a. Longshore Bars . The term bar has been applied to a number of 
quite different coastal features, including barrier islands (the "off- 
shore bar" of Johnson, 1919), ridge-and-runnel systems, and linear shoals. 
(Duane, et al., 1972.) Longshore bars are unrelated to any of these fea- 
tures. They appear to most nearly resemble a ridge-and-runnel system, 
but differ in that longshore bars are located at the breaker position and, 
at least in part, are eroded out of the bottom by the falling breaker, 
whereas the ridge-and-runnel system is an accretionary feature migrating 
landward across the surf zone. 

The typical longshore bar, as described from the observations of 
Shepard (1950) and the experiments of Keulegan (1948) , is a ridge of sand 
parallel to the shore and formed at the breaking position of large plung- 
ing breakers. Longshore bars seem most directly related to the height of 
larger breakers (not necessarily of maximum height). The depth to the sea- 
ward bar increases with the height of the larger breakers along the Pacific 
coast. (Shepard, 1950.) Bars form readily in tidal seas, but seem better 



4-78 



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4-79 



developed in tideless seas such as Lake Michigan (Fig. 4-27). (Saylor 
and Hands, 1970.) Keulegan (1948) found that the ratio of depth of long- 
shore trough to depth of bar was approximately 1.69 in laboratory experi- 
ments, but most field measurements showed less depth difference, averag- 
ing about 1.23 (based on MLLW) for 276 measurements from the Scripps pier. 
(Shepard, 1950.) According to Shepard, bars are not significant on slopes 
steeper than 4° (1 on 14). 

There is evidence that longshore bars, as described, are formed by a 
transient condition when waves of a given height and period plunge on a 
relatively plane sand slope. Shepard found that steep storm waves elimi- 
nate the bars rather than build them, and that bars form after the largest 
storm wave subsided. Such a relation is consistent with plunging breaker 
conditions predicted by the breaker type parameter (Galvin, 1972). Be- 
cause bars are formed by high waves, they may persist through long inter- 
vals of low waves. Once formed, bars may trigger the breaking of higher 
waves, dissipating the wave energy and thus reducing beach erosion. (Davis 
and Fox, 1972; and Zwambom, Fromme, and Fitzpatrick, 1970.) 

Laboratory observations show that longshore bars form when waves 
plunge, and that these bars are absent when waves spill. With constant 
wave conditions, a wave may plunge initially on a steep sand slope and 
form a bar. The beach then erodes forming a flatter slope, which changes 
the breaker type to spilling, which eliminates the pronounced longshore 
bar. For other constant wave conditions, a wave may spill initially on 
a gentle sand slope, and no pronounced bar forms. Later in the test, the 
breaker position migrates closer to the steeper foreshore, the breaker 
begins to plunge, and a longshore bar forms. 

b. Steepness Effect . The distinction between profiles with pro- 
nounced berms (usually without bars) and profiles with eroded foreshores 
(often with longshore bars) is well known. (Nayak, 1970, and Johnson, 
1956.) Early laboratory results suggested that the shape of the profile 
depends on deepwater steepness, H^/Lo, of the waves reaching the beach. 
Between 1936 and 1956, laboratory experiments were made which led to the 
assumption that beach profiles generally eroded if Hq/^o exceeded 0.025 
and accreted if ho/ho was less than about 0.02. 

However, neither field data nor prototype-size laboratory experiments 
support this widely used criterion. (Saville, 1957.) Field and prototype- 
size laboratory data of Saville showed that beaches eroded at signifi- 
cantly lower deepwater steepness than the value of 0.025 derived from 
model laboratory experiments. Saville (1957) concluded that the absolute 
size of wave height was probably as important as steepness in determining 
the profile. 



4-80 



c. Dimensionless Fall Time . Prediction of accreted (berm) or erod- 
ed (bar) profiles is possible using a dimensionless fall-time parameter 

Ho 

F^ = , (4-20) 

O VrT 

where H^ is deepwater wave height, T is wave period, and Vj? is the 
fall velocity of the beach sediment. (Dean, 1973.) 

The fall-time parameter Vq, is plotted against deepwater steep- 
ness, Ho/Lo, in Figure 4-29 using the profile data of Rector (1954, Table 
1, Column 25) and Saville (1957, unpublished). These data include wave 
heights ranging from about 0.05 foot to 5.0 feet, a range compatible with 
field conditions. They also include a range of initial slopes. 

In Figure 4-29, the line of demarcation between deposition offshore 
and deposition onshore is approximately at the value F^ = 1. More com- 
plete separation is possible when F^ is plotted against H^/Mj. (See 
Figure 4-30.) 

Values of F^ > 1 indicate that the time required by the particle 
to fall a distance about equal to the maximum depth in the surf zone is 
greater than the time available between arriving wave crests. Thus, 
values of F,^ significantly greater than 1 suggest significant concen- 
trations of suspended sediment, which are expected to diffuse seaward 
and deposit offshore. 

Since values of V^ range from about 0.066 foot per second (2 centi- 
meters per second) for fine sand to 0.49 foot per second (15 centimeters 
per second) for coarse sand (Fig. 4-31), ¥q ranges from about 0.25 for 
low swell on coarse sand beaches to 10 or more for storm waves on fine 
sand beaches. Such values suggest that typical field and laboratory con- 
ditions define a range of conditions within which the importance of sus- 
pended sediment in the surf zone may vary from significant to negligible. 

The effect of temperature on fall velocity (Fig. 4-31) is important 
enough to be critical under some conditions. It appears that temperature 
by itself, in its effect on fall velocity, can change a profile from erod- 
ing to accreting. 

To summarize results on berm-bar criteria, the dimensionless fall 
time, Vq = H^/(Vy?T), Equation 4-20 provides an estimate of the separa- 
tion between berm and bar-type profiles. A value of V^ between 1 and 2 
appears to be the critical value, with F^ = 2 being more appropriate for 
prototype-size waves. For ¥q less than the critical value, the beach 
accretes above MLW. The effect of fall velocity is important. Deepwater 
wave steepness by itself is an unreliable criterion for prototype condi- 
tions. 

4-81 



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( Kohler and Galvin,l973) 

Deepwater Steepness, Hq/Lq 



Figure 4-29. Berm-Bar Criterion Based on Diraensionless Fall 
Time and Deep Water Steepness 



4-82 



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WaveHeight-to-Groin Size Ratio, Ho/M<, (ft/ft) 

Figure 4-30. Berm-Bar Criterion Based on Dimensionless Fall 
Time and Height-to-Grain Size Ratio 



4-83 



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4.526 Slope of the Foreshore . The foreshore is the steepest part of the 
beach profile. The equilibrium slope of the foreshore is a useful design 
parameter, since this slope, along with the berm elevation, determines 
minimum beach width. 

The slope of the foreshore tends to increase as the grain size in- 
creases. (U.S. Army, Beach Erosion Board, 1933; Bascom, 1951; and King, 
1972, p. 324.) This relationship between size and slope is modified by- 
exposure to different wave conditions (Bascom, 1951; and Johnson, 1956); 
by specific gravity of beach materials (Nayak, 1970; and Dubois, 1972); 
by porosity and permeability of beach material (Savage, 1958); and prob- 
ably by the tidal range at the beach. Analysis by King (1972, p. 330) 
suggests that slope depends dominantly on sand size, and also signifi- 
cantly on an unspecified measure of wave energy. 

Figure 4-32 shows trends relating slope of the foreshore to grain 
size along the Florida Panhandle, New Jersey-North Carolina, and the 
U.S. Pacific coasts. Trends shown on the figure are simplifications of 
actual data, which are plotted in Figure 4-33. The trends show that, 
for constant sand size, slope of the foreshore usually has a low value 
on Pacific beaches, intermediate value on Atlantic beaches, and high 
value on Gulf beache*:. 

This variation in foreshore slope from one region to another appears 
to be related to the mean nearshore wave heights. (See Figures 4-10, 4-11, 
and Table 4-4.) The gentler slopes occur on coasts with higher waves. An 
increase in slope with decrease in wave activity is illustrated by data 
from Half Moon Bay (Bascom, 1951), and is indicated by the results of King 
(1972, p. 332). 

The inverse relation between slope and wave height is partly caused 
by the relative frequency of steep or high eroding waves which produce 
gentle foreshore slopes and low accretionary post-storm waves which pro- 
duce steeper beaches. (See Figures 4-1, 4-26, and 4-2 7.) 

The relation between foreshore slope and grain size shows greater 
scatter in the laboratory than in the field. However, the tendency for 
slope of the foreshore to increase with decreasing mean wave height is 
supported by laboratory data of Rector (1954, Table 1). In this labora- 
tory data, there is an even stronger inverse relation between deepwater 
steepness, Wq/Lo, and slope of the foreshore than between Wq a^d the 
slope. 

To summarize the results on foreshore slope for design purposes, the 
following statements are supported by available data: 

(a) Slope of the foreshore on open sand beaches depends principally 
on grain size, and (to a lesser extend) on nearshore wave height. 

(b) Slope of the foreshore tends to increase with increasing median 
grain size, but there is significant scatter in the data. 



4-85 



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4-87 



(c) Slope of the foreshore tends to decrease with increasing wave 
height, again with scatter. 

(d) For design of beach profiles on ocean or gulf beaches, use 
Figure 4-32, keeping in mind the large scatter in the basic data on Fig- 
ure 4-33, much of which is caused by the need to adjust the data to 
account for differences in nearshore wave climate. 

4.53 LONGSHORE TRANSPORT RATE 

4.531 Definitions and Methods . Littoral drift is the sediment (usually 
sand) moved in the littoral zone imder action of waves and currents. The 
rate, Q, at which littoral drift is moved parallel to the shoreline is 
the longshore transport rate. Since this movement is parallel to the 
shoreline, there are two possible directions of motion, right and left, 
relative to an observer standing on the shore looking out to sea. Move- 
ment from the observer's right to his left is motion toward the left, in- 
dicated by the subscript It. Movement toward the observer's right is 
indicated by the subscript rt. 

Gross longshore transport rate, Qg, is the sum of the amounts of 
littoral drift transported to the right and to the left, past a point on 
the shoreline in a given time period. 

Qg = Qrt + Qfit ' (4-21) 

Similarly, net longshore transport rate, Q^, is defined as the 
difference between the amounts of littoral drift transported to the right 
and to the left past a point on the shoreline in a given time period. 

%=%t- Q£. • C4-22) 

The quantities 0^.^, Q£t> On ^^id Q^ have engineering uses: for 
example, Qg is used to predict shoaling rates in uncontrolled inlets; 
Qj^ is used for design of protected inlets and for predicting beach ero- 
sion on an open coast; Q^,^ and Q^^ are used for design of jetties and 
impoundment basins behind weir jetties. In addition, Q^ provides an 
upper limit on other quantities. 



Occasionally, the ratio 



7 = — , (4-23) 

^1 



rt 

is known, rather than the separate values Q^^ and Qj,^. Then Q^ is 
related to Q^ in terms of y by ^ 

(1 +7) 
Q„ = Q„ ^^ — ■ (4-24) 

This equation is not very useful when y approaches 1. 

4-88 



Longshore transport rates are usually given in units of volume per 
time (cubic yards per year in the U.S.). Typical rates for oceanfront 
beaches range from 10^ to 10^ cubic yards per year. (See Table 4-6.) 
These volume rates typically include about 40 percent voids and 60 per- 
cent solids. 

At present, there are four basic methods to use for the prediction 
of longshore transport rate: 

1. The best way to predict longshore transport at a site is to 
adopt the best known rate from a nearby site, with modifictions based 
on local conditions. 

2. If rates from nearby sites are unknown, the next best way to 
predict transport rates at a site is to compute them from data showing 
historical changes in the topography of the littoral zone (charts, sur- 
veys, and dredging records are primary sources). 

3. If neither Method 1 nor Method 2 is practical, then it is 
accepted practice to use either measured or calculated wave conditions 
to compute a longshore component of "wave energy flux" which is related 
through an empirical curve to longshore transport rate. (Das, 1972.) 

4. A recently developed empirical method (Galvin, 1972) is avail- 
able to estimate gross longshore transport rate from mean annual near- 
shore breaker height. The gross rate, so obtained, can be used as an 
upper limit on net longshore transport rate. 

Method 1 depends largely on engineering judgement and local data. 
Method 2 is an application of historical data, which gives usable answers 
if the basic data are reliable and available at reasonable cost, and the 
interpretation is based on a thorough knowledge of the locality. By 
choosing only a few representative wave conditions, I-tethod 3 can usually 
supply an answer with less work than Method 2, but with correspondingly 
less certainty. Because calculation of wave statistics in Method 3 follows 
an established routine, it is often easier to use than researching the 
hydrographic records and computing the changes necessary for Method 2. 
Method 4 requires mean nearshore breaker height data. Section 4.532 
utilizes Methods 3 and 4. Methods 1 and 2 are discussed in Section 4.8. 

4.532 Energy Flux Method . Method 3 is based on the assumption that long- 
shore transport rate, Q, depends on the longshore component of energy flux 
in the surf zone. The longshore energy flux in the surf zone is approxi- 
mated by assuming conservation of energy flux in shoaling waves, using 
small -amplitude theory, and then evaluating the energy flux relation at 
the breaker position. The energy flux per unit length of wave crest, or, 
equivalently, the rate at which wave energy is transmitted across a plane 
of unit width perpendicular to the direction of wave advance is (from 
Section 2.238, combining Equations 2-39 and 2-40): 

P = EC = ^ H2 C . (4-25) 



4-89 



Table 4-6. Longshore 


Transport Rates 


from U.S. Coasts 


Location 


Predominant 

Direction of 

Transport 


Longshore* 
Transport 
(cu.yd./yr.) 


Date of 
Record 


Reference 


Atlantic Coast 


Suffolk County, N.Y. 
Sandy Hook, N.J. 
Sandy Hook, N.J. 
Asbury Park, N.J. 
Shark River, N.J. 
Manasquan, N.J. 
Barneget Inlet, N.J. 
Absecon Inlet, N.J. t 
Ocean City, N.J. t 
Cold Spring Inlet, N.J. 
Ocean City, Md. 
Atlantic Beach, N.C. 
Hillsboro Inlet, Fla. 
Palm Beach, Fla. 


W 
N 
N 
N 
N 
N 
S 

s 
s 
s 
s 

E 
S 

s 


200,000 
493,000 
436,000 
200,000 
300,000 
360,000 
250,000 
400,000 
400,000 
200,000 
150,000 
29,500 
75,000 
150,000 

to 
225,000 


1946-55 

1885-1933 

1933-51 

1922-25 

1947-53 

1930-31 

1939-41 

1935-46 

1935-46 

1934-36 
1850-1908 
1850-1908 
1925-30 


U.S. Army (1955a) 
U.S. Army (1954b) 
U.S. Army (1954b) 
U.S. Army (1954b) 
U.S. Army (1954b) 
U.S. Army (1954b) 
U.S. Army (1954b) 
U.S. Army (1954b) 
U.S. Congress (1953a) 
U.S. Congress (1953b) 
U.S. Army (1948a) 
U.S. Congress (1948) 
U.S. Army (1955b) 
U.S. Army (1947) 


Gulf of Mexico 


Pinellas County, Fla. 
Perdido Pass, Ala. 


S 
W 


50,000 
200,000 


1922-50 
1934-53 


U.S. Congress (1954a) 
U.S. Army (1954c) 


Pacific Coast 


Santa Barbara, Calif. 
Oxnard Plain Shore, Calif. 
Port Hueneme, Calif. 
Santa Monica, Calif. 
El Segundo, Calif. 
Redondo Beach, Calif. 
Anaheim Bay, Calif, t 
Camp Pendleton, Calif. 


E 

S 

s 
s 
s 

s 

E 
S 


280,000 
1,000,000 
500,000 
270,000 
162,000 
30,000 
150,000 
100,000 


1932-51 
1938-48 

1936-40 
1936-40 

1937-48 
1950-52 


Johnson (1953) 
U.S. Congress (1953d) 
U.S. Congress (1954b) 
U.S. Army (1948b) 
U.S. Army (1948b) 
U.S. Army (1948b) 
U.S. Congress (1954c) 
U.S. Army (1953a) 


Great Lakes 


Milwaukee County, Wis. 
Racine County, Wis. 
Kenosha, Wis. 
HI. State Line to Waukegan 
Waukegan to Evanston, 111. 
South of Evanston, 111. 


S 
S 
S 
S 

s 
s 


8,000 
40,000 
15,000 
90,000 
57,000 
40,000 


1894-1912 

1912-49 

1872-1909 


U.S. Congress (1946) 
U.S. Congress (1953e) 
U.S. Army (1953b) 
U.S. Congress (1953f) 
U.S. Congress (1953f) 
U.S. Congress (1953f) 


Hawaii 


Waikiki Beach t 


~ 


10,000 




U.S. Congress (1953g) | 



(from Wiegel, 1964; Johnson, 1957) 
* Transport rates are estimated net transport rates, Qn. In some cases, these approximate the gross transport rates, Qg. 

t Method of measurement is by Accretion except for Absecon Inlet, and Ocean City, New Jersey, and Anaheim Bay, 
California, by Erosion, and Waikiki Beach, Hawaii, by Suspended Load Samples. 



4-90 



If the wave crests make an angle, a with the shoreline, the energy flux 
in the direction of wave advance per unit length of beaah is 

PgH^ 
P cos a = — — C cos a , (4-26) 

8 * 

and the longshore component is given by 

— Pg 1 

?£ = P cos a sin a = -^ H C cos a sin a , 

8 * 

or since cos a sin a = 1/2 sin 2a 



Pp = 77 H^ C sin 2a . (4-27) 

* lo * 



The surf-zone approximation of Pj^, is written as P^.^. 

Pfi. = ll^S '^2ai, . (4-28) 

Usable formulations of this surf- zone approximation can be obtained by 
several methods. The principal approximation is in evaluating C^ and H 
at the breaker position. It is standard practice to approximate the group 
velocity, Cg, by the phase velocity, C, at breaking. The phase velo- 
city may then be approximated by either linear wave theory (Equation 2-3) 
or by solitary wave theory (Equation 4-13). 

Figure 4-34 presents the longshore component of wave energy flux in 
a dimensionless form, P^/pg^ H? T, as a function of breaker steepness, 

Hi/gT^, and the angle the wave crest makes with the shoreline in either 
deep water a^, or at the breaker line, ai^. Figure 4-34 is based on 
Equation 4-28 using linear wave theory to determine C^ and assuming that 
refraction is by straight parallel bottom contours. Figure 4-35 can be 
used to determine the longshore component of wave energy flux when breaker 
height, H^,, period, T, and angle, aj,, are known- -for example, for 
surf observation data. The use of Figure 4-34 is illustrated by an 
example problem below. 



For linear theory, in shallow water, C^ « C and 

P«. = ^ Hfc C sin 2aj, , (4-29) 

where H^j and a^ are the wave height and direction at breaking and C 
is the wave speed from Equation 2-3, evaluated in a depth equal to 1.28 
Hi. 



4-91 




4" 6" 8' 10' 20° 

Oeepwater Wave Angle, Qo, degrees 



60° 80° 100° 



Figure 4-34. Longshore Component of Wave Energy Flux in Dimensionless Form 
as a Function of Breaker Conditions 



4-92 



************** EXAMPLE PROBLEM 



*************** 



GIVEN : A breaking wave with height, Hj, = 4 feet; period T = 7 seconds. 
Surf observations indicate that the wave crest at breaking makes an 
angle, ajy = 6° with the shoreline. 

FIND ; 

(a) The longshore component of wave energy flux. 

(b) The angle the wave made with the shoreline when it was in deep 
water, Uq. 

SOLUTION : Calculate 

Hfe 4.0 

-— = r = 0.00254 . 

gT^ 32.2 (7.0)2 

Enter Figure 4-34 to the point where the line labeled aj, = 6" crosses 
H2,/gT2 = 0.00254 and read Pji/pg^ "f "^ = 6.7 x lO"** from the left axis 
and a(-) = 18° from the bottom axis. 

The longshore energy flux can then be calculated as, 

Pg = 0.00067 pg2 Hi T , 

Pg = 0.00067 (2) (32.2)2 (4 q)2 (70) . 

Pg = 155.6, say 160 ft.-lb./ft.-sec. , 

and 

«o = 18° • 

************************************* 

For offshore conditions, the group velocity is equal to one-half the 
deepwater wave speed C^, where C^ is given by Equation 2-7, and the 
refraction coefficient, K^, can be determined by the methods of Section 
2.32. Hence, 

P£s = ^T(H^K^)2 3i„2a, . (4-30) 

Figure 4-35 also presents the longshore component of wave energy flux 
in a dimensionless form, P£,/(pg^ H^ T), as a function of deepwater wave steep- 
ness, H^/gT^, and the angle the wave crest makes with the shoreline in 
either deep water, a^, or at the breaker line, a^,. Refraction by 
straight parallel bottom contours is again assumed. As illustrated by 

4-93 




E 
o 
o 



ID 



(i,°H,H/''d 



4-94 



the example problem below. Figure 4-35 can be used to determine the 
longshore component of wave energy flux when deepwater wave height, H^, 
period, T, and deepwater wave angle, a^, are known. 



************** EXAIIPLE PROBLEM 



*************** 



GIVEN : A wave in deep water has a height, H^ = 5 feet and a period, 
T = 7 seconds. While in deep water, the wave crest makes an angle. 



a^ = 25 with the shoreline. 



FIND: 



(a) The longshore component of wave energy flux. 

(b) The angle the wave makes with the shoreline when it breaks [assum- 
ing refraction is by straight, parallel bottom contours). 



SOLUTION: Calculate, 



Ho 



5.0 

= 0.0032 



gT^ 32.2 (7.0) 

Enter Figure 4-35 with a^, = 25° and H^/gT^ = 0.0032, read P^/pg^H^ T = 
1.5 X 10"^. This corresponds with a breaker angle a^, = 9.5° which is 
obtained by interpolation between the dashed curves of constant aj,. 
Tlierefore, 

Pj = (0.0015)pg^H2T , 

Pj = (0.0015) (2) (32.2)2 (5)2 (7 q) , 

Pj = 544.3 , say 540 ft.-lb./ft.-sec. , 
and 

afc = 9.5° . 
************************************** 

Equations 4-25 and 4-28 are valid only if there is a single wave 
train with one period and one height. However, most ocean wave condi- 
tions are characterized by a variety of heights with a distribution usu- 
ally described by a Rayleigh distribution. (See Section 3.22.) For a 
Rayleigh distribution, the correct height to use in Equation 4-28 or in 
the formulas shown in Table 4-8 is the root-mean-square amplitude. How- 
ever, most wave data are available as significant heights, and coastal 
engineers are used to dealing with significant heights. 

Significant height is implied in all equations for P^g. The value 
of Pj^g computed using significant height is approximately twice the 



4-95 



value of the exact energy flux for sinusoidal wave heights with a Rayleigh 
distribution. Since this means that P^g is proportional to energy flux 
and not equal to it, P^s is referred to as the longshore energy flux 
factor in Table 4-8 and the following sections. 

Longshore energy flux in this general case (P^) is given by equa- 
tion 4-27. This is an exact equation for the longshore component of 
energy flux in a single small -amplitude, periodic wave. This equation 
is good for any specified depth, but because the wave refracts, P^ will 
have different values as the wave moves into shallower water. The value 
of Pj, in Equation 4-27 can be manipulated through use of small -an^Jlitude 
wave theory to obtain the four equivalent formulas for Pjj, shown in Table 
4-7. 

In order to use Pj, for longshore transport computations in the surf 
zone, it is necessary to approximate P^ for conditions at the breaker 
position. These approximations are shown as P^g in Table 4-8, evaluated 
in foot-pound per second units. The bases for these approximations are 
shown in Table 4-9. Measurements show that the longshore transport rate 
depends on P^g. (See Figures 4-36 and 4-37.) 

As implied by the definition of Pusy the energy flux factors in 
Figures 4-36 and 4-37 are based on significant wave heights. The plotted 
P£g values were obtained in the following manner. For the field data of 
Watts (1953) and Caldwell (1956) , the original references give energy flux 
factors based on significant height, and these original data (after unit 
conversion) are plotted as Pj^g in Figures 4-36 and 4-37. Similarly, the 
one field point of Moore and Cole (1960), as adopted by Saville (1962), 
is assumed to be based on significant height. (See Figure 4-37.) Finally, 
the field data of Komar (1969), are given in terms of root-mean-square 
energy flux. This energy flux is multiplied by a factor of 2 (Das, 1972), 
converted to consistent units, and then plotted in Figure 4-36 and 4-37. 

For laboratory conditions (Fig. 4-36 only), waves of constant height 
are assumed. When these heights are used in the equations of Table 4-8, 
the result is an approximation of the exact longshore energy flux. In 
order to plot the laboratory data in terms of an energy-flux factor con- 
sistent with the plotted field data, this energy flux is multiplied by 2 
before plotting in Figure 4-36. 

For the purpose of this section, it is assumed that the shoaling co- 
efficient. Kg, for nearshore breaking waves is equivalent to the breaker 
height index, H^^/Ho', found from observation. (See Figure 2-65.) 

The choice of equations to determine P^g depends on the data avail- 
able. The right hand columns of Tables 4-7 and 4-8 tabulate the data 
required to use each of the formulas. An example using the second P^g 
formula is given in Section 4.533. 

Possible changes in wave height due to energy losses as waves travel 
over the Continental Shelf are not considered in these equations. Such 



4-96 



Table 4-7. Longshore Energy Flux.Pg , for a Single Periodic Wave in 
Any Specified Depth. (Four Equivalent Expressions from 
Small-Amplitude Theory) 



Equation 


(energy /time/distance) 


Data Required 
(any consistent units) 


4-31 
4-32 
4-33 
4-34 


2C^(%Esin2a) 
C (74 1^ sin 2q^) 

^R^o ('/^ ^0 si" 2a) 
(2C)(K|C^)-»C^C/4Esin2a^) 


d,T,H,a^ 
d, T, H^.tto 

d.T.H.a^.a 



no subscript indicates a variable at the specified depth 

where small-ampUtude theory is valid 
Cg = group velocity (see assumption lb, Table 4-9) 
C^ = deepwater 

d = water depth 

H = significant wave height 

T = wave period 

a = angle between wave crest and shoreline 



K^ — refraction coefficient 



/ cos Ug ' 

\ cos a 



Table 4-8. Approximate Formulas for Computing Longshore 
Energy Flux Factor, Pp , Entering the Surf Zone 



Equation 


(ft. -lbs. /sec. /ft. of beach front) 


Data Required 

(ft.-sec. units) 


4-35 


32.1 Hj, 5/2 sin2aj, 


"fc'"b 


4-36 


18.3 H^ 5/2 (cos a^)!''* sin 2a^ 


Hc."o 


4-37 


20.5 T H^ sin ay cos a^ 


T.H^> ««."?, 


4-38 


100.6 (H^/T) sin a^ 


T,Hj,,a, 



H^ = deepwater 

Hy = breaker position 

H = significant wave height 

T = wave period 

a = angle between wave crest and shoreline 

See Table 4-7 for equivalent small amplitude equations and 
Table 4-9 for assumptions used in deriving Pj from Pp. 



4-97 



Table 4-9. Assumptions for Pj Formulas in Table 4-8. 



1. Formula 1 — Equation 4-35 

a. Energy density at breaking is given by linear theory, 

E = (pg4)/8 « SHg 

b. Group velocity equals wave speed at breaking, and breaking speed is given by 
solitary wave theory according to the approximation. (Galvin, 1967, Equation 11.) 



C^ = C«(2gHj,)^/^= 8.02 (Hfe)^/^ 
c. a can be replaced by ay 

2. Formula 2 — Equation 4-36 

a. Same as lb above 

b. H, is related to H by refraction and shoaling coefficients, where the coefficients 
are evaluated at the breaker position 

c. Refraction coefficient K^^ given by small-amplitude theory; shoaUng coefficient 
K assumed constant, so that 

d. (Hj,)^/^ = 1.14 (cosaj'/^ h/^ 
if (cosaj,)'/^ = 1.0 

and (Kj'/^ = 1.14 

3. Formula 3 — Equation 4-37 

a. Refraction coefficient at breaking is given by small-amplitude theory. 

4. Formula 4 — Equation 4-38 

a. Same as la above 

b. Same as lb above 

c. Same as 3a above 

d. Cos ttjj = 1.0 

NOTE: Constants evaluated for foot-pound-second units. Small-amplitude theory is assumed valid in 
deep water. Nearshore contours are assumed to be straight and parallel to the shoreline. 

4-98 



10' 



10' 



o 

O 

ce 

rio^ 

o 

a. 



^10^ 

.c 

V) 

o> 

c 
o 



10' 



10 



10" 






t • 

I • 






■,0 



o 



For Design, use Figure 4-37 



O t^; 

°^ ° ^r °%° 

O O Oo o 

<b oo ^ 

o ° o o 



• Field Doto, Significont Height 

° Loborofory Dato, Quartz Sand, Periodic Woves 



10 



-1 



10 



10' 



10" 



0' 



Longshore Energy Flux Factor, Pa, ft.-lbs/sec/ft. of beach front 



Figure 4-36. Longshore Transport Rate Versus Energy Flux Factor for Field 
and Lab Conditions 



4-99 



10^ 



a. 

ro 
T5 



10* 



o 



O 



o 

(A 



.c 
I/) 

c 
o 



10' 









...^^ 


-r- +-'--:t 




gp2H 


^.*fe* 


2s-m44ss 


r=si 


H 


^ 




ZiH 




-- 


; ■ r: ; :i; :"i ■■ : . '""t 


■■ 












. .-i 










J 








ij 


; • : 1 : - -. , i^Y' ', ■.:- 


:--: 
























1 






-- - . 


/ 




:■ 


--— ■ 










-_._- 


~ 


















/- 


1 




4^9 


1 1 Mo/e 


nc) 


^ 


1 - 


— 


— 


-:■-=■ 


--~^— ^ 




i 'i , 


:: 


-■-- 


y 


<<^ 










;:; 


];. 




p.4 q; Caidwe 


ii 




i -; 


■ 




;:.;j. 


---T 


-— 


-_;-_-:- 


::..:: 


ftr 


•■> 




^' 


r-/: 


^ 


--:--- 


1 


"y - ^igriiti'cofiT'fff 
: Po\ is )t&ine 




rrti 

rom 


lised in 
lable^4 




i«i 


/:: :; 


"P 


f^ 


;:::,;. 


- '!. ;-]-! ! 




^ 








■" 
















■ : 




: " 




/^ 




o 


.. : 




" 






■ ■ ■ - . ■ i, -■ i" :' '-■■ ■ ' 














—-■ 












- 1 


■ P 










■ . ■ : ;: ;:"l :,, : r 


: 








-^■ 


■ 




/ O 


















. . :;|:^:: ^.:! '.i ^ - 










- " 1 





" 


r'r- 


:s:.:^2:.. 


"Ttrr 


TiTr 












-- 


^.■■l'i-—--n— 


^___ 




■ "■ --*—}-—— --- 




_ J 


w 


1 




--^T^ 


--"t 


— - 








r- 


„j:.; 


-- 






ifi :'..'■ 








:::;|:: 


fe' 


.: , . 




■1^ 








i::^-^^:-: — --^ 


i^ 


'■'■ ■ '- 


:;^±ES:^ 


d 


i^ 


f 




— 


^ 




+ -- 


— - 


™ 






"- 


..'::::• 


-- 


, ' ~ ___ — -_ — _ 






-:^ 


^i 


X 


rr 






■J S 


-Eoii;(i?n 


— i — 1 

.■ - 1 , 


j^: 


1^ 


ts 






' 




^ '1 


lux 




































ir ^ 


















"■.] ■ 


. , 












'1^^ 






-. i.- 








- ■ : . ; 


::;:]; :. 


i' 


). • 








J_... .1:^^:.- 


— 


r-- 


:; ^^:^ 


,: 


---j. 


— 


— 








:H^ 




-i- 

■■ ■-!■■ 

■■ 






-f 


- -t r /-— ^ 


?^ 


^\ 


i' 


^ "^ 




"- 


^~ — 


— - 










-- 


— t- 


■ ■ . - 


-■■ 


[ 




' : 














1 


liiiii- ^- 


-^ 


:i-z: 




■-;- — H 


1 


Hi 


~y^v. 


^ 


; "1 ir , 


— ■ ■ \-Tr- 


:X-tVf. 

i 




jii' 


:: 


■ : ■ - 
, ; 

"Tt;' 


x: 


^-:- 


t 



I 10 10' 

Longshore Energy Flux Factor, P£5, Ft -lbs/sec/ ft of beacti front 



10" 



Figure 4-37. Design Curve for Longshore Transport Rate versus Energy 
Flux Factor. Only field data are included. 



4-100 



changes may reduce the value of P^g when deepwater wave height statis- 
tics are used as a starting point for computing P^g. (Walton, 1972; 
Bretschneider and Reid, 1954; and Bretschneider, 1954.) 

The Equations 4-35 through 4-38 in Table 4-8 are related to the Equa- 
tion for Eq previously recommended for use with this method (Caldwell, 
1956, Equations 5 and 6; or the equations in Figure 2-22, page 175, CERC 
Technical Report No. 4, 1966 edition) by a constant 

E^ = (8.64 X 10'*)P£^ (4-39) 

where E^ is in units of foot-pounds per foot per day and P^g is in 
units of foot-pounds per foot per second. 

The term in parenthesis for Equation 4-32 in Table 4-7 is identical 
to the longshore force of Longuet-Higgins (1970a). This longshore force 
also correlates well with the longshore transport rate. 

The relation between Q and P^g in Figures 4-36 and 4-37 can be 
approximated by 

Q = (7.5X103)Pj^ (4-40) 

Equation 4-40 tends to overestimate Q at the higher values of P^g 
for the plotted field data, but it falls below the estimated rates com- 
puted from the data of Johnson (1952). (See Das, 1972, Figure 6.) The 
value of 7.5 x 10^ in Equation 4-40 is approximately twice the equiva- 
lent value from the design curve of CERC Technical Report No. 4, 1966 
edition, and is about 5 percent greater than the value estimated by Komar 
and Inman (1970). 

Judgement is required in applying Equation 4-40. Although the data 
in Figures 4-36 and 4-37 appear to follow a smooth trend, the log- log 
scale compresses the data scatter. For example, the average difference 
between the plotted points from field data and the prediction given by 
Equation 4-40 is at least 28 percent of the value of prediction (average 
difference derived by Das (1972) is 42 percent). In addition, some in- 
complete measurements suggest transport rates ranging from tv;o orders of 
magnitude below the line (Thornton, 1969) to one order of magnitude above 
the line, (Johnson, 1952.) These additional data are plotted by Das 
(1972). 

As an aid to computation. Figures 4-38 and 4-39 give lines of con- 
stant Q based on Equation 4-40 and Equations 4-35 and 4-36 for Pis 
given in Table 4-8. To use Figures 4-38 and 4-39 to obtain the longshore 



4-101 



transport rate, only the (H^j, a^) data and Figure 4-38, or the (H^, a^) 
data and Figure 4-39 are needed. If the shoaling coefficient is signifi- 
cantly different from 1.3, multiply the Q obtained from Figure 4-39 by 
the factor 0.88 vlCg. (See Table 4-9, Assumption 2d.) 

Figure 4-39 applies accurately only if o.q is a point value. If a^ 
is a range of values, for example a 45° sector implied by the direction 
NE, then the transport evaluated from Figure 4-39 using a single value 
of a^ for NE may be 12 percent higher than the value obtained by aver- 
aging over the 45° sector implied by NE. The more accurate approach is 
given in the example problem of the next section. 

The imit for Q is a volume of deposited quartz sand (including 
voids in the volume) per year. Bagnold (1963) suggests using immersed 
weight instead of volume in the unit for longshore transport rate (Section 
4.521), since immersed weight is the pertinent physical variable related 
to the wave action causing the sediment transport. Use of an immersed 
weight unit does eliminate the difference between lightweight material 
and quartz that occurs if volume units are used. (Das, 1972.) However, 
in coastal engineering design, it is the volume and not the immersed 
weight of eroded or deposited sand that is important, and since beach 
sand is predominantly quartz (specific gravity 2.65), volume is directly 
proportional to immersed weight. On some beaches the sand may be calcium 
carbonate which has a specific gravity ranging from 2.87 (calcite) to 
2.98 (aragonite) in pure form. Naturally occurring oolitic aragonite 
sand with a specific gravity of 2.88 (Monroe, 1969), has an immersed 
weight 14 percent greater than pure quartz sand. Since the longshore 
transport Equation 4-40, with one exception (Watts, 1953; and Das, 1971, 
p. 14) , is based on quartz sand, then oolitic sand beaches may have 
slightly lower longshore transport rates than is suggested by comparison 
with data from quartz sand beaches. However, the scatter in the data 
(Fig. 4-36) makes such a specific gravity effect difficult to detect. 

4.533 Energy Flux Example (Method 3) . Assume that an estimate of the 
longshore transport rate is required for a locality on the north-south 
coastline along the west edge of an inland sea. The locality is in an 
area where stronger winds blow out of the northwest and north, resulting 
in a deepwater distribution of height and direction as listed in Table 
4-10o Assume the statistics were obtained from visual observations 
collected over a 2-year interval at a point 2 miles offshore by seamen 
aboard vessels entering and leaving a port in the vicinity. This type 
of problem, based on SSMO wave statistics (Section 4.34), is discussed 
in detail by Walton, (1972), and Walton and Dean (1973). Shipboard 
data are subject to uncertainty in their applicability to littoral trans- 
port, but often they are the only data available. It is assumed that 
shipboard visual observations are equivalent to significant heights. 
(Cartwright, 1972; and Walton, 1972.) 



4-102 



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4-103 




Qo iQO 20° 30° 40° 50° 60° 70° 80° 90° 
Deepwater Angle, Of q > ( degrees ) 
( Do not use averaged angle ) 

Figure 4-39. Longshore Transport Rate as a Function of Deepwater 
Height and Deepwater Angle 



4-104 



Table 4-10. Deepwater Wave Heights, in Percent by Direction, off East-Facing 
Coast of Inland Sea 



Compass Direction 


N 


NE 


E 


SE 


S 


Other* 


ao 


90° 


45° 


0" 


-45° 


-90° 




H,t(ft.) 


9 


10 


6 


5 


5 




1 


2 


5 


5 


2 


2 


2 




3 


4 


3 


1 


1 


1 




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2 


1 










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8 


1 












Total 


22 


19 


9 


8 


8 


34 



*Calm conditions, or waves from SW, W, or NW. 

t Shipboard visual observations assumed equivalent to significant height 
(See Walton, 1972.) 

This problem could be solved using Figure 4-39, but for illustration, 
and because of a slightly higher degree of accuracy from the direction 
data given, the problem is illustrated in detail. 

In this example, the available data are the joint frequency distri- 
bution of H^ and a^. For each combination of a.^ and H^, the 

corresponding CL h is calculated for Table 4-11 in the following 

o ' o 
manner. The basic equation is a form of Equation 4-40 written 



'% ' ^o 



- |f^^^Ja..H > 



(4-41) 



where f is the decimal frequency, which is the percent frequency in 
Table 4-10, divided by 100. The constant. A, is of the type used in 
Equation 4-40. 

Since the available data are a^ and H(p, the appropriate equation 
for ViQ is given in Table 4-8. If A = 7.5 x 10^ as in Equation 4-40 
and Equation 4-36 in Table 4-8 are used. 



where 



^%-Ho = 1-^73 X 10^ fH//^ F(,^| 



(4-42) 



(4-43) 



Tills direction term, f^(a^) , requires careful consideration. A 
compass point direction for the given data (Table 4-10) represents a 45' 



4-105 



sector of wave directions. If ^(.a^) is evaluated at a^ = 45° (NE or 
SE in the example problem), it will have a value 12 percent higher than 
the average value for FCa^) over 45° sector bisected by the NE or SE 
directions. Thus, if the data warrant a high degree of accuracy. Equa- 
tion 4-43 should be averaged by integrating over the sector of directions 
involved. 



Table 4-11. Computed Longshore 


Transport for East-Facing Coast of Inland Sea 


Qttp , Ho i"^ cu.yd./yr. from Equation 4-42 


Ho 


N* 


NE 


E 


SE 


S* 


1 


1.61 X lO^t 


11.26 X 103 ^ 


±1.52 X 103 


- 5.63 X 103 


-0.89 X 103 


2 


5.05 X 10^ 


31.84 X 103 


± 2.87 X 103 


-12.74 X 103 


-2.02X 103 


3 


11.13 X 10^ 


52.42 X 103 


± 3.96 X 103 


- 17.55 X 103 


-2.78 X 103 


4 


11.42 X 10^ 


36.03 X 103 








5 


9.98 X 103 










8 


32.31 X 103 










Totals 


71.50 X 103 


131.78 X 103 


(±8.35 X 103) 
16.70 X 103 


- 35.92 X 103 


-5.69 X 103 



% = 



(71.50 + 131.78 + 8.35) X 103 = 212 X 103 or 212,000 cu.yd./yr. 
(8.35 + 35.92 + 5.69) X 103 = 50 X 103 or 50,000 cu.yd./yr. 
Qrf ~ Qfif = 212 X 103 - 50 X 103 - 162 X 103 or 162,000 cu.yd./yr. 
Qft "*" Qfif "" 212 X 103 4- 50 X 103 = 262 X 103 or 262,000 cu.yd./yr. 



*Coast runs N-S so frequencies of waves from N and S are halved. 
tCalculation of this number is showrn in detail in the text. 



If F(a^) is evaluated at 



(waves from the E in the example 



problem), then F(a^) = 0. Actually, a^ = is only the center of a 
45° sector which can be expected to produce transport in both directions. 
Therefore, F(oio) should be averaged over 0° to 22.5° and 0° to -22.5°, 
giving F(a^) = ± 0.370 rather than 0. The + or - sign comes out of the 
sin 2a.Q term in F(ao) (Equation 4-43), which is defined such that trans- 
port to the right is positive, as implied by Equation 4-22. 

A further complication in direction data is that waves from the north 
and south sectors include waves traveling in the offshore direction. It 
is assumed that, for such sectors, frequency must be multiplied by the 
fraction of the sector including landward traveling waves. For example 
the frequencies from N and S in Table 4-10 are multiplied by 0.5 to 
obtain the transport values listed in Table 4-11. 



4-106 



To illustrate how values of Qa^' Ho listed in Table 4-11 were 
calculated, the value of Qa^, H^ ■'■^ here calculated for H^ = 1 and 
the north direction, the top value in the first column on Table 4-11. 
The direction term, FCa^) , is averaged over the sector from a = 67.5° 
to a = 90°, i.e., from NNE to N in the example. The average value of 
F(aci) is found to be 0.261. H^ to the 5/2 power is simply 1 for this 
case. The frequency given in Table 4-10 for H^^ = 1 and direction = 
north (NW to NE) is 9 percent or in decimal terms, 0.09. This is mul- 
tiplied by 0.5 to obtain the part of shoreward directed waves from the 
north sector (i.e., N to NE) resulting in f = 0.09 (0.5) = 0.045. Put- 
ting all these values into Equation 4-42 gives 

Qf^j = 1.373 X 10^ (0.045) (1)^2 (0.261) 

= 1,610 yd.Vyr. (See Table 4-11) 

Table 4-11 indicates the importance of rare high waves in determin- 
ing the longshore transport rate. In the example, shoreward moving 8- foot 
waves occur only 0.5 percent of the time, but they account for 12 percent 
of the gross longshore transport rate. (See Table 4-11.) 

Any calculation of longshore transport rate is an estimate of poten- 
tial longshore transport rate. If sand on the beach is limited in quan- 
tity, then calculated rates may indicate more sand transport than there 
is sand available. Similarly, if sand is abundant, but the shore is 
covered with ice for 2 months of the year, then calculated transport 
rates must be adjusted accordingly. 

The procedure used in this example problem is approximate and 
limited by the data available. Equation 4-42, and the other approxi- 
mations listed in Table 4-11, can be refined if better data are avail- 
able. An extensive discussion of this type of calculation is given by 
Walton (1972). 

Although this example is based on shipboard visual observations of 
the SSMO type (Section 4.34), the same approach can be followed with 
deepwater data from other sources, if the joint distribution of height 
and direction is known. At this level of approximation, the wave period 
has little effect on the calculation, and the need for it is bypassed as 
long as the shoaling coefficient (or breaker height index) reasonably 
satisfies the relation (Kg)^/^ = 1.14. (See Assumption 2d, Table 4-9.) 
For waves on sandy coasts, this relation is reasonably satisfied. (e.g., 
Bigelow and Edmondson, 1947, Table 33; and Goda, 1970, Figure 7.) 

4.534 Enyirical Prediction of Gross Longshore Transport Rate (Method 4) . 
Longshore transport rate depends partly on breaker height, since as 
breaker height increases, more energy is delivered to the surf zone. At 
the same time, as breaker height increases, breaker position moves off- 
shore widening the surf zone and increasing the cross-section area through 
which sediment moves. 

4-107 



Galvin (1972) showed that when field values of longshore transport 
rate are plotted against mean annual breaker height from the same locality, 
a curve 

Q = 2 X 10= H^ , (4-44) 

forms an envelope above almost all known pairs of (Q, Hj,) , as shown in 
Figure 4-40. Here, as before, Q is given in units of cubic yards per 
year; Hjy in feet. 

Figure 4-40 includes all known (Q, Hj,) pairs for which both Q and 
H2j are based on at least 1 year of data, and for which Q is considered 
to be the gross longshore transport rate, Q^, defined by Equation 4-21. 
Since all other known (Q, H^,) pairs plot below the line given by Equation 
4-41, the line provides an upper limit on the estimate of longshore trans- 
port rate. From the defining equations for Q^ and Q^, any line that 
forms an upper limit to longshore transport rate must be the gross trans- 
port rate, since the quantities Qrt> Q£t> ^^^ Qn» ^^ defined in Section 
4.531, are always less than or equal to Q^^. 

In Equation 4-44, wave height is the only independent variable, and 
the physical explanation assumes that waves are the predominant cause of 
transport. (Galvin, 1972.) Therefore, where tide-induced currents or 
other processes contribute significantly to longshore transport. Equation 
4-44 would not be the appropriate approximation. The corrections due to 
currents may either add or subtract from the estimate of Equation 4-44, 
depending on whether currents act with or against prevailing wave-induced 
transport. 

4,535 Method 4 Example (Empirical Prediction of Gross Longshore Transport 
Rate. Near the site of the problem outlined in Section 4.533, it is de- 
sired to build a small craft harbor. The plans call for an unprotected 
harbor entrance, and it is required to estimate costs of maintenance 
dredging in the harbor entrance. The gross transport rate is a first 
estimate of the maintenance dredging required, since transport from 
either direction could be trapped in the dredged channel. Wave height 
statistics were obtained from a wave gage in 12 feet of water at the end 
of a pier. (See Columns (1) and (2) of Table 4-12.) Heights are avail- 
able as empirically determined significant heights. (Thompson and Harris, 
1972.) (To facilitate comparison, the frequencies are identical to the 
deepwater frequencies of onshore waves in Table 4-10 for the problem of 
Section 4.533. That is, the frequency associated with each H„ in Table 
4-12 is the sum of the frequencies of the shoreward Hq on the correspond- 
ing line of Table 4-10.) 

The breaker height, H^j, in the empirical Equation 4-44 is related 
to the gage height, Hg, by a shoaling-coefficient ratio, (Ks)^/(Kg)^, 
where (Kg)^, is the shoaling coefficient (Equation 2-44), (H/H^ in 



4-108 



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:::;:;::.::, ::::.;ij 





0.1 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 

Mean Breaker Height, Hb, (feet ) (Gaivin,i972) 

Figure 4-40. Upper Limit on Longshore Transport Rates 



4-109 



Table 4-12. Example Estimate of Gross Longshore Transport Rate for Shore of Inland Sea 



(1) 


(2) 
f 


(3) 


(4) 


(5) 
(3) X (4) 


(ft.) 




(ft.) 




(ft.) 


1 


0.28 


0.28 


1.2 


0.336 


2 


0.125 


0.25 


1.2 


0.300 


3 


0.075 


0.225 


1.2 


0.270 


4 


0.02 


0.08 


1.1 


0.088 


5 


0.005 


0.025 


1.1 


0.028 


8 


0.005 


0.04 


1.0 


0.040 








Hi 


= 1.062 



(1) H^ = 

(2) f = 



(4) 



(^^l 



significant height reduced from gage records, assumed to correspond 
to the height obtained by visual observers 

decimal frequency of wave heights 
= assumed shoaling-coefficient ratio. 



(5) Hj, = Z — -fH^ = 1.062ft. 

Q = 2 X 10^ Hg = 2.26 X 10^ cu.yd./yr. from Equation 4^4 
Note that shoreward-moving waves exist only 51 percent of the time. 



4-110 



Table C-1 Appendix C) evaluated at the breaker position and i^s^g i^ 
the shoaling coefficient evaluated at the wave gage: 

H. - H (4-45) 

Kg or H/H^ can be evaluated from small -amplitude theory, if wave-period 
information is available from the wave gage statistics. For simplicity, 
assume shoaling-coefficient ratios as listed in Column 4 of Table 4-12. 
Such shoaling coefficient ratios are consistent with the shoaling co- 
efficient of Kg=1.3 (between deepwater and breaker conditions) assumed 
in deriving P^g (Table 4-9) , and with the fact that waves on the inland 
sea of the problem would usually be steep, locally generated waves. 

Column 5 of the table is the product fHg (Kg)^/(Kg)g. The sum 
(1.06 feet) of entries in this column is assumed equivalent to the aver- 
age of visually observed breaker heights. Substituting this value in 
Equation 4-44, the estimated gross longshore transport rate is 226,000 
cubic yards per year. It is instructive to compare this value with the 
value of 262,000 cubic yards per year obtained from the deepwater example. 
(See Table 4-11.) The two estimates are not expected to be the same, 
since the same wave statistics have been used for deep water in the first 
problem and for a 12-foot depth in the second problem. However, the numer- 
ical values do not differ greatly. It should be noted that the empirical 
estimate just obtained is completely independent of the longshore energy 
flux estimate of the deepwater example. 

In this example, wave gage statistics have been used for illustrative 
purposes. However, visual observations of breakers, such as those listed 
in Table 4-4, would be even more appropriate since Equation 4-44 has been 
"calibrated" for such observations. On the other hand, hindcast statis- 
tics would be less satisfactory than gage statistics due to the uncertain 
effect of nearshore topography on the transformation of deepwater statis- 
tics to breaker conditions. 

4.6 ROLE OF FOREDUNES IN SHORE PROCESSES 

4.61 BACKGROUND 

The cross section of a barrier island shaped solely by marine hydrau- 
lic forces has three distinct subaerial features: beach, crest of island, 
and deflation plain. (See Figure 4-41.) The dimensions and shape of the 
beach change in response to varying wave and tidal conditions (Section 
4.524), but usually the beach face slopes upward to the island crest - the 
highest point on the barrier island cross section. From the island crest, 
the back of the island slopes gently across the deflation plain to the 
edge of the lagoon separating the barrier island from the mainland. These 
three features are usually present on duneless barrier island cross sec- 
tions; however, their dimensions may vary. 



4-III 




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4-112 



Island crest elevation is determined by the nature of the sand form- 
ing the beach, and by the waves and water levels of the ocean. The beach 
and waves interact to determine the elevation of the limit of wave runup - 
the primary factor in determining island crest elevation. Normally the 
island crest elevation is almost constant over long sections of beach. 
However, duneless barrier island crest elevations vary with geographical 
area. For example, the crest elevation typical of Core Banks, North 
Carolina, is about +6 feet MSL; +4 feet MSL is typical for Padre Island, 
Texas; +11 feet MSL is typical for Nauset Beach, Massachusetts. 

Landward of the upper limit of wave uprush or berm crest are the back- 
shore and the deflation plain. This area is shaped by the wind, and in- 
frequently by the flow of water down the plain when the island crest is 
overtopped by waves. (e.g., Godfrey, 1972.) Obstructions which trap 
wind-transported sand cause the formation of dunes in this area. (See 
discussion in Section 6.4 Sand Dunes.) Beachgrasses which trap wind- 
transported sand from the beach and the deflation plain are the major 
agent in creating and maintaining foredunes. 

4.62 ROLE OF FOREDUNES 

Foredunes, the line of dunes just behind a beach, have two primary 
functions in shore processes. First, they prevent overtopping of the 
island during some abnormal sea conditions. Second, they serve as a 
reservoir for beach sand. 

4.621 Prevention of Overtopping . By preventing water from overtopping, 
foredunes prevent wave and water damage to installations landward of the 
dune. They also block the water transport of sand from the beach area to 
the back of the island and the flow (overwash) of overtopping sea water. 

Large reductions in water overtopping are effected by small increases 
in foredune crest elevations. For example, the hypothetical 4-foot dune 
shown in Figure 4-41 raises the maximum island elevation about 3 feet to 
an elevation of 6 feet. On this beach of Padre Island, Texas, the water 
levels and wave runup maintain an island crest elevation of +4 feet MSL 
(about 2 feet above MHW) . This would imply that the limit of wave runup 
in this area is 2 feet (the island crest elevation of 4 feet minus the 
MHW of 2 feet). Assuming the wave runup to be the same for all water 
levels, the 4-foot dune would prevent significant overtopping at water 
levels up to 4 ft MSL (the 6-foot effective island height at the dune 
crest minus 2 feet for wave runup). This water level occurs on the aver- 
age once each 5 years along this section of coast. (See Figure 4-42.) 
Thus, even a low dune, one which can be built with vegetation and sand 
fences in this area in 1 year (Woodard et al . , 1971) provides consider- 
able protection against wave overtopping. (See Section 5.3 and 6.3.) 

Foredunes or other continuous obstructions on barrier islands may 
cause unacceptable ponding from the land side of the island when the la- 
goon between the island and mainland is large enough to support the needed 
wind setup. (See Section 3.8.) There is little danger of flooding from 



4-113 



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4-114 



this source if the lagoon is less than 5 miles wide. Where the lagoon 
is wider (especially 10 miles or greater) flooding from the lagoon side by 
wind setup should be investigated before large dune construction projects 
are undertaken. 

4.622 Reservoir of Beach Sand . During storms, erosion of the beach occurs 
and the shoreline recedes. If the storm is severe, waves attack and erode 
the foredunes and supply sand to the beach; in later erosion stages, sand 
is supplied to the back of the island by overwash. (Godfrey, 1972.) 

Volumes of sand eroded from beaches during storms have been estimated 
in recent beach investigations. Everts (1973) reported on two storms dur- 
ing February 1972 which affected Jones Beach, New York. The first storm 
eroded an average of 27,000 cubic yards per mile above mean sea level for 
the 9-mile study area; the second storm (2 weeks later) eroded an average 
of 35,000 cubic yards of sand per mile above mean sea level at the same 
site. Losses at individual profiles ranged up to 120,000 cubic yards per 
mile. Davis (1972) reported a beach erosion rate on Mustang Island, Texas, 
following Hurricane Fern (September 1971), of 12.3 cubic yards per linear 
foot of beach for a 1,500-foot stretch of beach (about 65,000 cubic yards 
per mile of beach). On Lake Michigan in July 1969, a storm eroded an aver- 
age of 3.6 cubic yards per linear foot of beach (about 29,000 cubic yards 
per mile) from an 800-foot beach near Stevensville, Michigan. (Fox, 1970.) 
Because much of the eroded sand is usually returned to the beach by wave 
action soon after the storm, these volumes are probably representative of 
temporary storm losses. 

Volumes equivalent to those eroded during storms have been trapped 
and stored in foredunes adjacent to the beach. Foredunes constructed 
along Padre Island, Texas, and Core Banks, North Carolina, (Section 6.43 
and 6.447) contain from 30,000 to 80,000 cubic yards of sand per mile of 
beach. Assuming the present rate of entrapment of sand continues for the 
next 3 years at these sites, sand volumes ranging from 50,000 to 160,000 
cubic yards per mile of beach will be available to nourish eroding beaches 
during a major storm. Sand volumes trapped during a 30-year period by 
European beachgrass at Clatsop Spit, Oregon, averaged about 800,000 cubic 
yards per mile of beach. Tlius, within a few years, foredunes can trap and 
store a volume of sand equivalent to the volumes eroded from beaches dur- 
ing storms of moderate intensity. 

4.623 Long-Term Effects . Dolan, (1972-73) advances the concept that a 
massive, unbroken foredune line restricts the landward edge of the surf 
zone during storms causing narrower beaches and thus increased turbulence 
in the surf zone. The increased turbulence causes higher sand grain attri- 
tion and winnowing rates and leads to accelerated losses of fine sand, an 
erosive process that may be detrimental to the long-range stability of bar- 
rier islands. However, as discussed in Section 4.521, the effects of sedi- 
ment size are usually of secondary importance in littoral transport pro- 
cesses - processes which are important in barrier island stability. In 
addition, geographical location is probably more important in determining 
beach sand size than dune effects, since both fine and coarse sand beaches 



4-115 



front major foredune systems in different geographical locations. For 
example, fine sand beaches front a massive foredune system on Mustang 
Island, Texas, and coarse sand beaches front dunes on the Cape Cod spits, 

Godfrey (1972) discusses the effect of a foredune system on the 
long term stability of the barrier islands of the Cape Hatteras and Cape 
Lookout National Seashores, North Carolina. Important implicit asstimp- 
tions of the discussion are that no new supply or inadequate new supplies 
of sand are available to the barrier island system, and that rising sea 
level is, in effect, creating a sand deficit by drowning some of the 
available island volume. The point of the geomorphic discussion is that 
under such conditions the islands must migrate landward to survive. A 
process called "oceanic overwash" (the washing of sand from low foredunes 
or from the beach over the island crest onto the deflation plain by over- 
topping waves) is described as an important process in the landward migra- 
tion of the islands. Since a foredune system blocks overtopping and pre- 
vents oceanic overwash, foredunes are viewed as a threat to barrier island 
stability. 

Granted the implicit assumptions and a geologic time frame, the 
geomorphic concept presented has convincing logic and probably has merit. 
However, the assumptions are not valid on all barrier islands or at all 
locations in most barrier islands or at all locations in most barrier 
island systems. Too, most coastal engineering projects are based on 
a useful life of 100 years or less. In such a short period, geologic 
processes, such as sea-level rise, have a minor effect in comparison with 
the rapid changes caused by wind and waves. Therefore, the island crest 
elevation and foredune system will maintain their elevation relative to 
the mean water level on stable or accreting shores over the life of most 
projects. On eroding shores, the foredunes will eventually be eroded 
and overwash will result in shoreward migration of the island profile; 
sand burial and wave and water damage will occur behind the original 
duneline. Therefore, planning for and evaluation of the probable suc- 
cess of a foredune system must consider the general level of the area of 
the deflation plain to be protected, the rate of sea level rise, and the 
rate of beach recession. 

4.7 SEDIMENT BUDGET 
4.71 INTRODUCTION 

4.711 Sediment Budget . A sediment budget is based on sediment removal, 
transportation and deposition, and the resulting excesses or deficiencies 
of material quantities. Usually, the sediment quantities are listed 
according to the sources, sinks, and processes causing the additions and 
subtractions. In this chapter, the sediment is usually sand, and the 
processes are either littoral processes or the changes made by man. 

The purpose of a sediment budget is to assist the coastal engineer 
by: identifying relevant processes; estimating volume rates required for 
design purposes; singling out significant processes for special attention; 
and, on occasion, through balancing sand gains against losses, checking 
the accuracy and completeness of the design budget. 



4-116 



Sediment budget studies have been presented by Johnson (1959), Bowen 
and Inman (1966), Vallianos (1970), Pierce (1969), and Caldwell (1966). 

4.712 Elements of Sediment Budget . Any process that increases the 
quantity of sand in a defined control volume is called a sauroe. Any 
process that decreases the quantity of sand in the control volume is 
called a sink. Usually, sources are identified as positive and sinks 
as negative. Some processes (longshore transport is the most important) 
function both as source and sink for the control volume, and these are 
called conveoting processes. 

Point sources or point sinks are sources or sinks that add or sub- 
tract sand across a limited part of a control volume boundary. A tidal 
inlet often functions as a point sink. Point sources or sinks are gener- 
ally measured in units of volume per year. 

Line sources or tine sinks are sources or sinks that add or subtract 
sand across an extended segment of a control volume boundary. Wind trans- 
port landward from the beaches of a low barrier island is a line sink for 
the ocean beach. Line sources or sinks are generally measured in units of 
volume per year per lonit length of shoreline. To compute the total effect 
of a line source or sink, it is necessary to multiply this quantity by the 
total length of shoreline over which the line source or sink operates. 

The following conventions are used for elements of the sediment 
budget: 

Q^ is a point source 
Q^ is a point sink 
q^ is a line source 

q^ is a line sink 

These subscripted elements of the sediment budget are identified by name 
in Table 4-13 according to whether the element makes a point or line con- 
tribution to the littoral zone, and according to the boundary across which 
the contribution enters or leaves. Each of the elements is discussed in 
following sections. 

The length of shoreline over which a line source is active is in- 
dicated by hi and the total contribution of the line source or line 

*+ * — 

sink by Q^ or Q^ , so that in general 

Q* = b,. q,. . (4-46) 

4-117 



Table 4-13. Classification of Elements in the Littoral Zone Sediment Budget 



Location of 
Source or Sink 


Offshore Side of 
Littoral Zone 


Onshore Side of 
Littoral Zone 


Within 
Littoral Zone 


Longshore Ends 

of 

Littoral Zone 


Point Source 
(cu.yd./yr.) 


offshore shoal or 
island 


Rivers, streams* 


Replenishment 


Longshore 
transport in* 


Point Sink 
(cu.yd./yr.) 


Submarine canyon 


Inlets* 


Mining, extractive 
dredging 


Q4 

Longshore 

transport out* 


Line Source 
(cu.yd./yr./ft. of beach) 


+ 

Sand transport 
from the offshore 


Coastal erosion 

including erosion 

of dunes and cliffs* 


Beach erosion* 
CaCO, production 




Line Sink 
(cu.yd./yr./ft. of beach) 


Sand transport to 
the offshore 


Overwash 

Coastal land and 

dune storage 


Beach storage* 
CaCOg losses 





*Naturally occurring sources and sinks that usually are major elements in the sediment budget. 

It is often useful to specify a source or sink as a fraction, k^, 
of the gross longshore transport rate: 



Q,- = k . Q. 



s 



(4-47) 



In a complete sediment budget, the difference between the sand added 
by all sources and the sand removed by all sinks should be zero. In the 
usual case, a sand budget calculation is made to estimate an unknown ero- 
sion or deposition rate. This estimated rate will be the difference result- 
ing from equating known sources and sinks. The total budget is shown sche- 
matically as follows: 

Sum of Sources - Sum of Sinks = 0, or 

Sum of Known Sources - Sum of Known Sinks = Unknown (Sought) 

Source or Sink 




= 



(4-48) 



The Q| are obtained using Equation 4-46 and the appropriate q^ and 
hi. The subscript, i, equals 1, 2, 3, or 4 and corresponds to the sub- 
scripts in Table 4-13. 



4-118 



4.713 Sediment Budget Boundaries . A sediment budget is used to identify 
and quantify the sources and sinks that are active in a specified area. 
By so doing, erosion or deposition rates are determined as the balance 
of known sinks and sources. Boundaries for the sediment budget are deter- 
mined by the area imder study, the time scale of interest, and study pur- 
poses. In a given study area, adjacent sand budget compartments (control 
volumes) may be needed with shore-perpendicular boundaries at significant 
changes in the littoral system. For example, compartment boundaries may 
be needed at inlets, between eroding and stable beach segments, and between 
stable and accreting beach segments. Shore-parallel boundaries are needed 
on both the seaward and landward sides of the control volumes. They may be 
established wherever needed, but the seaward boundary is usually established 
at or beyond the limit of active sediment movement, and the landward bound- 
ary beyond the erosion limit anticipated for the life of the study. The 
bottom surface of a control volume should pass below the sediment layer 
that is actively moving, and the top boundary should include the highest 
surface elevation in the control volume. Thus, the budget of a particular 
beach and nearshore zone would have shore parallel boundaries landward of 
the line of expected erosion and at or beyond the seaward limit of signifi- 
cant transport. A budget for barrier island sand dunes might have a bound- 
ary at the bay side of the island and the landward edge of the backshore. 

A sediment budget example and analysis are shown in Figure 4-43. 
This example considers a shoreline segment along which the incident wave 
climate can transport more material entering from updrift. Therefore 
the longshore transport in the segment is being fed by a continuously 
eroding sea cliff. The cliff is composed of 50 percent sand and 50 per- 
cent clay. The clay fraction is assumed to be lost offshore while the 
sand fraction feeds into the longshore transport. The budget balances 
the sources and sinks using the following continuity equation: 

Sum of Known Sources - Sum of Known Sinks = Difference 

An example calculation is shown in Figure 4-43. 

4.72 SOURCES OF LITTORAL MATERIAL 

4.721 Rivers . It is estimated that rivers of the world bring about 3.4 
cubic miles or 18.5 billion cubic yards of sediment to the coast each year 
(voliime of solids without voids). (Stoddard, 1969; from Strakhov, 1967.) 
Only a small percentage of this sediment is in the sand size range that is 
common on beaches. The large rivers which account for most of the volume 
of sediment carry relatively little sand. For example, it is estimated 
(Scruton, 1960) that the sediment load brought to the Gulf of Mexico each 
year by the Mississippi River consists of 50 percent clay, 48 percent silt, 
and only 2 percent sand. Even lower percentages of sand seem probable for 
other large river discharges. (See Gibbs, 1967, p. 1218, for information 
on the Amazon River.) But smaller rivers flowing through sandy drainage 
areas may carry 50 percent or more of sand. (Chow, 1964, p. 17-20.) In 
southern California, sand brought to the coast by the floods of small 
rivers is a significant source of littoral material. (Handin, 1951; and 
Norris, 1964.) 

4-119 



Offshore Sink 




4. Longshore Tronsporf 
Qa ■ 



IN 



WATER 



I Longshore Transport 

*- Q4 



r r 



Sediment Budget Boundary 




Top of Cliff 



LAND 



ERODING SHORELINE-PLAN VIEW 

(Not to Scole) 



Budget Boundory 

ri Recession 



\ \ 
I \ 

> \,*i-First Survey Profile 




v^ Vy- Second Survey Profile 




\ N. 




^\ \ MHW 




v_ J--- ^ 


MLW 



Seoword Limit of 
Active Erosion 



SECTION A-A 



(Not to Scale) 



Assumptions 



Budget Calculations 



Q4 = 100,000 yd.Vyr. 

q+ = 1 yd?/yr./U. 

q~ = 0.5yd.Vyr./{l. 

b = 10,000 ft. 



Sum of Sources — Sum of Sinks = Difference 



Find Q: 



(q; +q^b)-(Q;+q;b) = o 

(10^ + 1.0 X 10"*) - (Q~ + 0.5 X lO'*) = o 



Q^ = 110,000-5,000 



Q^ = 105,000 yd.Vyr. 
Figure 4-43. Basic Example of Sediment Budget 



4-120 



Most of the sediment carried to the coast by rivers is deposited 
in comparatively small areas, often in estuaries where the sediment is 
trapped before it reaches the coast. (Strakhov, 19670 The small frac- 
tion of sand in the total material brought to the coast and the local 
estuarine and deltaic depositional sites of this sediment suggest that 
rivers are not the immediate source of sediment on beaches for much of 
the world's coastline. Many sources of evidence indicate that sand- 
sized sediment is not supplied to the coasts by rivers on most segments 
of the U.S. Atlantic and Gulf coasts. Therefore, other sediment sources 
must be important. 

4.722 Erosion of Shores and Cliffs . Erosion of the nearshore bottom, 
the beach, and the seaward edge of dunes, cliffs, and mainland (Fig. 4-44) 
results in a sand loss. In many areas, erosion from cliffs of one area is 
the principal source of sand for downdrift beaches. Kuenen (1950) esti- 
mates that beach and cliff erosion along all coasts of the world totals 
about 0.03 cubic mile or 160 million cubic yards per year. Although this 
amount is only about 1 percent of the total solid material carried by 
rivers, it is a major source in terms of sand delivered to the beaches, 
since the sand fraction in the river sediments is usually small, and is 
usually trapped before it reaches the littoral zone. Shore erosion is an 
especially significant source where older coastal deposits are being ero- 
ded, since these usually contain a large fraction of sand. 

If an eroding shore maintains approximately the same profile above 
the seaward limit of significant transport while it erodes, then the ero- 
sion volume per foot of beach front is the vertical distance from dune 
base or berm crest to the depth of the seaward limit (h) , multiplied by 
the horizontal retreat of the profile. Ax. (See Figure 4-44.) 

Figure 4-44 shows three equivalent volumes, all indicating a net ero- 
sion of hAx„ To the right in Figure 4-44 is a typical beach profile. 
The dashed line profile below it is the same as the solid line profile. 
The horizontal distance between solid and dashed profiles is Ax, the 
horizontal retreat of the profile due to (assumed) uniform erosion. The 
unit volume loss, hAx, between dune base and depth to seaward limit is 
equivalent to the unit volume indicated by the slanted parallelogram on 
the middle of Figure 4-44. The unit volume of this parallelogram, hAx, 
is equivalent to the shaded rectangle on the left of Figure 4-44. If the 
vertical distance, h is 40 feet, and Ax = 1 foot of horizontal erosion, 
then the unit volume lost is 40/27, or 1.5 cubic yards per foot of beach 
front . 

4.723 Transport from Offshore Slope . An uncertain and potentially signifi- 
cant source in the sediment budget is the contribution from the offshore 
slopCo However, hydrography, sediment size distribution, and related evi- 
dence discussed in Section 4.523 indicate that contributions from the con- 
tinental shelf to the littoral zone are probably negligible in many areas. 
Most shoreward moving sediment appears to originate in areas fairly close 
to shoreo Significant onshore-offshore transport takes place within the 
littoral zone due to seasonal and storm-induced profile changes and to 



4-121 



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4-122 



erosion of the nearshore bottom and beaches, but in the control volume 
defined, this transport takes place within the control volume. Transport 
from the offshore has been treated as a line source. 

In some places, offshore islands or shoals may act as point sources 
of material for the littoral zone. For example, the drumlin islands and 
shoals in Boston Harbor and vicinity may be point sources for the nearby 
mainland. 

4.724 Windblown Sediment Sources . To make a net contribution to the 
littoral zone in the time frame being considered, windblown sand must 
come from a land source whose sand is not derived by intermediate steps 
from the same littoral zone. On U.S. ocean coasts, such windblown sand 
is not a significant source of littoral materials. Where wind is impor- 
tant in the sediment budget of the ocean shore, wind acts to take away 
sand rather than to add it, although local exceptions probably occur. 

However, windblown sand can be an important source, if the control 
volume being considered is a beach on the lagoon side of a barrier island. 
Such shores may receive large amounts of windblown sand. 

4.725 Carbonate Production . Dissolved calcium carbonate concentration 
in the ocean is near saturation, and it may be precipitated under favor- 
able conditions. In tropical areas, many beaches consist of calciiom car- 
bonate sands; in temperate zones, calcium carbonate may be a significant 
part of the littoral material. These calciiom carbonate materials are gen- 
erally fragments of shell material whose rate of production appears to in- 
crease with high temperature and with excessive evaporation. (See Hayes, 
1967.) Oolitic sands are a nonbiogenic chemical precipitate of calcium 
carbonate on many low latitude beaches. 

Quantitative estimates of the production of calcium carbonate sedi- 
ment are lacking, but maximum rates might be calculated from the density 
and rate of growth of the principal carbonate-producing organisms in an 
area. For example, following northeasters along the Atlantic coast of 
the U.S., the foreshore is occasionally covered with living clams thrown 
up by the storm from the nearshore zone. One estimate of the annual con- 
tribution to the littoral zone from such a source would assume an average 
shell thickness of about 0.04 foot completely covering a strip of beach 
100 feet wide all along the coast. On an annual basis, this would be 
about 0.15-cubic yard per year per foot of beach front. Such a quantity 
is negligible under almost all conditions. However, the dominance of 
carbonate sands in tropical littoral zones suggests that the rate of pro- 
duction can be much higher. 

4.726 Beach Replenishment . Beach protection projects often require 
placing sand on beaches. The quantity of sand placed on the beach in 
such beach-fill operations may be a major element in the local sediment 
budget. Data on beach-fill quantities may be available in Corps of Engi- 
neer District offices, in records of local government engineers, and in 
dredging company records. The exact computation of the quantity of a 



4-123 



beach fill is subject to uncertainties: the source of the dredged sand 
often contains significant but variable quantities of finer materials that 
are soon lost to the littoral zone; the surveys of both the borrow area 
and the replenished area are subject to uncertainty because sediment trans- 
port occurs during the dredging activities; and in practice only limited 
efforts are made to obtain estimates of the size distribution of fill 
placed on the beach. Thus, the resulting estimate of the quantity of 
suitable fill placed on the beach is uncertain. More frequent sampling 
and surveys could help identify this significant element in many sedi- 
ment budgets. 

4.73 SINKS FOR LITTORAL MATERIALS 

4.731 Inlets and Lagoons . Barrier islands are interrupted locally by 
inlets which may be kept open by tidal flow. A part of the sediment moved 
alongshore by wave action is moved into these inlets by tidal flow. Once 
inside the inlet, the sediment may deposit where it cannot be moved sea- 
ward by the ebb flow. (Brown, 1928.) The middleground shoals common to 
many inlets are such depositional features. Such deposition may be reduced 
when the ebb currents are stronger than the flood currents. (Johnson, 
1956.) 

It is evident from aerial photography (e.g., of Drum Inlet, N.C., 
Fig. 4-45) that inlets do trap significant quantities of sand. Caldwell's 
(1966) estimate of the sand budget for New Jersey, calculates that 23 
percent of the local gross longshore transport is trapped by the seven 
inlets in southern New Jersey, or about 250,000 cubic yards per year for 
each inlet. In a study of the south shore of Long Island, McCormick (1971) 
estimated from the growth of the f loodtide delta of Shinnecock Inlet (shown 
by aerial photos taken in 1955 and 1969) that this inlet trapped 60,000 
cubic yards per year. This amounts to about 20 percent of the net long- 
shore transport (Taney, 1961a, p. 46), and probably less than 10 percent 
of the gross transport. (Shinnecock Inlet is a relatively small inlet.) 
It appears that the rate at which an inlet traps sediment is higher imme- 
diately after the inlet opens than it is later in its history. 

4.732 Overwash . On low barrier islands, sand may be removed from the 
beach and dune area by overwashing during storms. Such rates may average 
locally up to 1 cubic yard per year per foot. Data presented by Pierce 
(1969) suggest that for over half of the shoreline between Cape Hatteras 
and Cape Lookout, North Carolina, the short term loss due to overwash was 
0.6 cubic yard per year per foot of beach front. Figure 4-46 is an aerial 
view of overwash in the region studied by Pierce (1969). Overwash does 
not occur on all barrier islands, but if it does, it may function as a 
source for the beach on the lagoon side. 

4.733 Backshore and Dune Storage . Sand can be temporarily withdrawn 
from transport in the littoral zone as backshore deposits and dune areas 
along the shore. Depending on the frequency of severe storms, such sand 
may remain in storage for intervals ranging from months to years. Back- 
shore deposition can occur in hours or days by the action of waves after 



4-124 




Figure 4-45. Sediment Trapped Inside Old Drim Inlet, 
North Carolina 



4-125 




( I November 1971 ) 



Figure 4-46. Overwash on Portsmouth Island, North 
Carolina 



4-126 



storms. Dune deposits require longer to form - months or years - because 
wind transport usually moves material at a lesser rate than wave transport. 
I£ the immediate beach area is the control volume of interest, and budget 
calculations are made based on data taken just after a severe storm, allow- 
ance should be made in budget calculations for sand that will be stored in 
berms through natural wave action. (See Table 4-5.) 

4.734 Offshore Slopes . The offshore area is potentially an important 
sink for littoral material. Transport to the offshore is favored by: 
storm waves which stir up sand, particularly when onshore winds create 

a seaward return flow; turbulent mixing along the sediment concentration 
gradient which exists between the sediment-water mixture of the surf zone 
and the clear water offshore; and the slight offshore component of gravity 
which acts on both the individual sediment particles and on the sediment- 
water mixture. 

It is often assumed that the sediment sorting toss that commonly 
reduces the volume of newly placed beach fill is lost to the offshore 
slopes. (Corps of Engineers, Wilmington District, 1970; and Watts, 
1956.) A major loss to the offshore zone occurs where spits build into 
deep water in the longshore direction. Sandy Hook, New Jersey, is an 
example. (See Figure 4-47.) It has been suggested (Bruun and Gerritsen, 
1959) that ebb flows from inlets may sometimes cause a loss of sand by 
jetting sediment seaward into the offshore zone. 

The calculation of quantities lost to the offshore zone is difficult, 
since it requires extensive, accurate, and costly surveys. Some data on 
offshore changes can be obtained by studies of sand level changes on rods 
imbedded in the sea floor (Inman and Rusnak, 1956), but without extending 
the survey beyond the boundary of the moving sand bed, it is difficult 
to determine net changes. 

4.735 Submarine Canyons . Probably the most frequently mentioned sinks 
for littoral materials are submarine canyons. Shepard (1963) and Shepard 
and Dill (1966) provide extensive description and discussion of the origin 
of submarine canyons. The relative importance of submarine canyons in 
sediment budgets is still largely unknown. 

Of 93 canyons tabulated by Shepard and Dill (1966), 34 appear to be 
receiving sediment from the coast, either by longshore transport or by 
transport from river mouths. Submarine canyons are thought to be espe- 
cially important as sinks off southern California. Herron and Harris 
(1966, p. 654) suggest that Mugu Canyon, California, traps about 1 mil- 
lion cubic yards per year of the local littoral drift. 

The exact mechanism of transport into these canyons is not clear, 
even for the La Jolla Canyon (California) which is stated to be the most 
extensively studied submarine feature in the world. (Shepard and Buff- 
ington, 1968.) Once inside the canyons, the sediment travels down the 
floors of the heads of the canyons, and is permanently lost to the litto- 
ral zone. 



4-127 




( 14 September 1969) 



Figure 4-47. Growth of a Spit into Deep Water, Sandy- 
Hook, New Jersey 



4-128 



4.736 Deflation . The loose sand that forms beaches is available to be 
transported by wind. After a storm, shells and other objects are often 
found perched on pedestals of sand left standing after the wind eroded 
less protected sand in the neighborhood. Such erosion over the total 
beach surface can amount to significant quantities. Unstabilized dunes 
may form and migrate landward, resulting in an important net loss to the 
littoral zone. Examples include some dunes along the Oregon coast (Coo- 
per, 1958), between Pismo Beach and Point Arguello, California (Bowen 
and Inman, 1966); central Padre Island (Watson, 1971); and near Cape Hen- 
lopen, Delaware (Kraft, 1971). Typical rates of transport due to wind 
range from 1 to 10 cubic yards per year per foot of beach front where 
wind transport is noticeable. (Cooper, 1958; Bowen and Inman, 1966; 
Savage and Woodhouse, 1968; and Gage, 1970.) However average rates prob- 
ably range from 1 to 3 cubic yards per year per foot. 

The largest wind-transported losses are usually associated with 
accreting beaches that provide a broad area of loose sand over a period 
of years. Sand migrating inland from Ten Mile River Beach in the vicin- 
ity of Laguna Point, California, is shown in Figure 4-48. 

Study of aerial photographs and field reconnaissance can easily 
establish whether or not important losses or gains from wind transport 
occur in a study area. However, detailed studies are usually required 
to establish the importance of wind transport in the sediment budget. 

4.737 Carbonate Loss . The abrasion resistance of carbonate materials 
is much lower than quartz, and the solubility of carbonate materials is 
usually much greater than quartz. However, there is insufficient evidence 
to show that significant quantities of carbonate sands are lost from the 
littoral zone in the time scale of engineering interest through either 
abrasion or solution. 

4.738 Mining and Dredging . From ancient times, sand and gravel have 
been mined along coasts. In some countries, for example Denmark and 
England, mining has occasionally had undesirable effects on coastal settle- 
ments in the vicinity. Sand mining in most places has been discouraged by 
legislation and the rising cost of coastal land, but it still is locally 
important. (Magoon, et al., 1972.) It is expected that mining will become 
more important in the offshore area in the future. (Duane, 1968, and Fisher, 
1969.) 

Such mining must be conducted far enough offshore so the mined pit 
will not act as a sink for littoral materials, or refract waves adversely, 
or substantially reduce the wave damping by bottom friction and percolation. 

Material is also lost to the littoral zone when dredged from naviga- 
ble waters (channels and entrances) within the littoral zone, and the 
dredged material is dumped in some area outside of the littoral zone. 
These dump areas can be for land fill, or in deep water offshore. This 
action has been a common practice, because the first costs for some 
dredging operations are cheaper when done this way. 



4-129 




(24 Moy 1972) 
Figure 4-48. Dunes Migrating Inland Near Laguna Point, California 



4-130 



4.74 CONVECTION OF LITTORAL MATERIALS 

Sources and sinks of littoral materials are those processes that 
result in net additions or net subtractions of material to the selected 
control volume. However, some processes may subtract at the same rate 
that they add material, resulting in no net change in the volume of lit- 
toral material of the control volume. 

The most important convecting process is longshore sediment transport. 
It is possible for straight exposed coastlines to have gross longshore 
transport rates of more than 1 million cubic yards per year. On a coast 
without structures, such a large Q^ can occur, and yet not be apparent 
because it causes no obvious beach changes. Other convecting processes 
that may produce large rates of sediment transport with little noticeable 
change include tidal flows, especially around inlets, wind transport in 
the longshore direction, and wave-induced currents in the offshore zone. 

Since any structure that interrupts the equilibrium convection of 
littoral materials will normally result in erosion or accretion, it is 
necessary that the sediment budget quantitatively identify all processes 
convecting sediment through the study area. This is most important on 
shores with high waves. 

4.75 RELATIVE CHANGE IN SEA LEVEL 

Relative changes in sea level may be caused by changes in sea level 
and changes in land level. Sea levels of the world are now generally ris- 
ing. The level of inland seas may either rise or fall, generally depend- 
ing on hydrologic influences. Land level may rise or fall due to tectonic 
forces, and land level may fall due to subsidence. It is often difficult 
to distinguish whether apparent changes in sea level are due to change in 
sea level, change in land level, or both. For this reason, the general 
process is referred to as relative change in sea level. 

While relative changes in sea level do not directly enter the sedi- 
ment budget process, the net effect of these elevation changes is to move 
the shoreline either landward (relative rise in sea level) or seaward 
(relative fall in sea level). It thus can result in the appearance of a 
gain or loss of sediment volume. 

The importance of relative change in sea level on coastal engineer- 
ing design depends on the time scale and the locality involved. Its 
effect should be determined on a case-by-case basis. 

4.76 SUMMARY OF SEDIMENT BUDGET 

Sources, sinks, and convective processes are summarized diagrammati- 
cally in Figure 4-49 and listed in Table 4-14. The range of contributions 
or losses from each of these elements is described in Table 4-14 measured 
as a fraction of the gross longshore transport rate, or as a rate given 
in cubic yards per year per foot of beach front. The relative importance 



4-131 



of elements in the sand budget varies with locality and with the bound- 
aries of the particular littoral control volume. (These elements are 
classified as point or line sources or sinks in Table 4-13, and the budget 
is summarized in Equation 4-48.) 



LAND 



Cliff, Dune and 
Backshore Erosion 

Beach Replenishment 

Rivers 

Dune and Backshore 
Storage 

Winds 
(supply and deflation) 



Inlets and Lagoons 
Including Overwash 

Mining 




OCEAN 



Offshore Slope 
(source or sink?) 

Submarine Canyon 



Dredging 



Calcium Carbonate 
(production and loss) 



Longshore Currents 
Tidal Currents 
Longshore Winds 

Figure 4-49. Materials Budget for the Littoral Zone 

In most localities, the gross longshore transport rate significantly 
exceeds other volume rates in the sediment budget, but if the beach is 
approximately in equilibrium, this may not be easily noticed. 

The erosion of beaches and cliffs and river contributions are the 
principal known natural sources of beach sand in most localities. Inlets, 



4-132 



Table 4-14. Sand Budget of the Littoral Zone 





Sources 


Rivers and streams 


The major source in the limited areas where rivers carry sand to the littoral 
zone. In affected areas notable floods may contribute several times Qg. 


Cliff, dune and backshore erosion 


Generally the major source where rivers are absent. 1 to 4 cu.yd./yr./ft. 


Transport from offshore 


Quantity uncertain. 


Wind transport 


Not generally important as a source. 


CaC03 production 


Significant in tropical climate. The value of 0.25 cu.yd./yr./ft. seems 
reasonable upper limit on temperate beach. 


Beach replenishment 


Varies from to greater than Qg. 





Sinks 


Inlets and lagoons 


May remove from 5 to 25 percent of Qg per inlet. Depends on number of 
inlets, inlet size, tidal flow characteristics, and inlet age. 


Overwash 


Less than 1 cu.yd./yr./ft. at most, and limited to low barrier islands. 


Beach storage 


Temporary, but possibly large, depending on beach condition when budget 
is made. (See Table 4-5, pages 4-72, 4-73.) 


Offshore slopes 


Uncertain quantity. May receive much fine material, some coarse material. 


Submarine canyons 


Where present, may intercept up to 80 percent of Qg. 


Deflation 


Usually less than 2 cu.yd./yr./ft. of beach front, but may range up to 10 
cu.yd./yr./ft. 


CaC03 loss 


Not known to be important. 


Mining and dredging 


May equal or exceed Qg in some localities. 



Convective Processes 



Longshore transport (waves) 



Tidal Currents 



Winds 



May result in accretion of Qg, erosion of Qj,, or no change depending on 
conditions of equilibrium. 

May be important at mouth of inlet and vicinity, and on irregular coasts 
with high tidal range. 

Longshore winds are probably not important, except in limited regions. 



4-133 



lagoons and deep water in the longshore direction comprise the principal 
known natural sinks for beach sand. Of potential, but usually unknown, 
importance as either a source or a sink is the offshore zone seaward of 
the beach. 

The works of man in beach replenishment and in mining or dredging 
may provide major sources or sinks in local areas. In a few U.S. locali- 
ties, submarine canyons or wind may provide major sinks, and calcium car- 
bonate production by organisms may be a major source. 

************** EXAMPLE PROBLEM *************** 

GIVEN: 



(a) An eroding beach 4.4 miles long at root of spit that is 10 miles 
long. Beaches on the remainder of the spit are stable. (See 
Figure 4-50a.) 

(b) A uniform recession rate of 3 feet per year along the eroding 
4o4 miles. 

(c) Depth of lowest shore parallel contour is -30 feet MSL, and 
average dune crest elevation is 15 feet MSL. 

(d) Sand is accumulating at the tip of the spit at an average rate of 
400,000 cubic yards per year. 

(e) The variation of y along the beaches of the spit is shown in 
Figure 4-51. (y = Q.it/^rt'' Equation 4-23.) 

(f) No sand accumulates to the right of the erosion area; no sand is 
lost to the offshore. 

(g) A medium-width jettied inlet is proposed which will breach the 
spit as shown in Figure 4-50a. 

(h) The proposed inlet is assumed to trap about 15 percent of the 
gross transport, Q^. 

(i) The 1.3-mile long beach to the right of the jettied inlet will 
stabilize (no erosion) and realign with y changing to 3.5. 

(j) The accumulation at the end of the spit will continue to grow 
at an average annual rate of 400,000 cubic yards per year after 
the proposed inlet is constructed. 

4-134 



(a) Site Sketch 



Reach 4 


Reach 3 




Reach 2 


Reach 1 


^ . 










Ocean 








C^^^Z^ 


fj 






Propo 


sad Inlet ' 



(b) Before Inlet 



Reach 4 


Reach 3 


Reach 2 


Reach 1 


Q^(4) = 400,000 
Q«(4) = 400,000 


Q^(3,4) = 490,000 
Q.(3.4) - 400,000 


Q^(2.3) = 530,000 
^Qn(2,3) = 318,000 


Q^(i,2) = 474,000 
%{i,2) - 284,000 


Q/t(4) " 400,000 


QrK3.4) ^ 45,000 
QiK3.4) - 445,000 


QrK2.3) ^ 105,000 
Q/f(2,3) ^ 425,000 


QrKi,2) ^ 95,000 
Q/Ki,2) = 379,000 











^(3)^3(3) = 82,000 b(2)q^(2) - 34,000 



(c) After Inlet 



Qn(4) 



400,000 
400,000 



Q^((4) = 400,000 



Q^(3,4) 
Qn(3,4) 



490,000 
400,000 



QrK3,4) 
9^f(3,4) 



45,000 
445,000 



Qn(2,3) 



Qrf(2,3) 
QiK2.3) 



512,000 
284,000 
114,000 
398,000 



^gU.2) 

Q«(l,2) 



^rt(l,2) 

^Mh2) 



474,000 
284,000 
95,000 
379,000 



LEGEND 



I } 



^(3)l3(3) " 193,000 Q" = 77,000 



(X Gross Volume 

Q^ Net Volume 
Q^^ Volume to Right 
Q^^ Volume to Left 

Q" Inlet Sink Volume 
bq^ Erosion Source Volume 

Figure 4-50. Summary of Example Problem Conditions and Results 



4-135 



10 - 



X 







Reach 4 
(average / = 9) 



Reaches 1-3 
(average / =4) 







10 



Distance from Spit Tip -Miles 



Figure 4-51, 



Variation of y with Distance Along the Spit, 
before Inlet Condition 



FIND: 



(a) Annual littoral drift trapped by inlet. 

(b) After-inlet erosion rate of the beach to the left of the inlet. 

(c) After-inlet nourishment needed to maintain the historic erosion 
rate on the beach to the left of the inlet. 

(d) After-inlet nourishment needed to eliminate erosion left of the 
inlet. 

SOLUTION : Divide the beach under study into four sand budget compart- 
ments (control volumes called reaches) as shown in Figure 4-50a. Shore 
perpendicular boundaries are established where important changes in the 
littoral system occur. To identify and quantify the hefore-inlet sys- 
tem, the continuity of the net transport rate along the spit must be 
established. The terminology of Figure 4-43 and Table 4-13 is used 



4-136 



for the sand budget calculation. The average annual volume of material 
contributed to the littoral system per foot of eroding beach reaches 
2 and 3 is: 

n+ - n+ - 1,A (15 + 30) (3) 

%(2) - ^5(5) -^-= Yl 

= 5.0 yd?/yryft. . 

Then, from Equation 4-46 the total annual contribution of the eroding 
beaches to the system can be determined as: 

^{(2) "*" ^1(3) " (1-3 f"i- + 3-1 "^i-) (5280 ft./mi.) (5 yd?/yr./ft.) 
= 1.16 X 105yd?/yr. 

Since there is no evidence of sand accumulation or erosion to the right 
of the eroding area, the eroding beach material effectively moves to the 
left becoming a component of the net transport volume (Q„) toward the 
end of the spit. Contiuity requires the erosion volume and Reach 1 Q^ 
must combine to equal the acretion at the end of the spit (400,000 cubic 
yards per year). Thus, Q^^ at the root of the spit is 

%(1 2) " 400,000 yd?/yr. - 116,000 yd?/yr. 

\{1,2)^ 284,000 yd.3/yr. 

Q^ across the boundary between Reaches 2 and 3 (Qj^r2 o)) is: 

%{2,3) = Q«(i,2) + \2) (15(2)) ^^ 

= 284,000 yd?/yr. + (1.3 mi.) 15,280— j (5 yd?/yr./ft.) 

= 318,000 yd.Vyr. 
Qyj across the boundary between Reaches 3 and 4 is: 

^n{3,4)^^n{2,3) + ^i) «(i)) 

= 318,000 yd?/yr. + (3.1 mi.) 15,280 — | (5 yd?/yr/ft.) 

= 4000,000 yd?/yr. 

This Qri(.3 h) moves left across reach 4 with no additions or subtrac- 
tions, and since the accretion rate at the end of the spit is 400,000 
cubic yards per year, the budget balances. Knowing Q^ and y for 
each reach, gross transport, Q^, transport to the right Q^,^ and 
transport to the left Qg^^ can be computed using the following equa- 
tions : 



4-137 



From Equation 4-24 



then 






Q.. = 



'^ 1+7 ' 



and 



%> Q-rt' ^lt> ^^^ %. ^°^ each reach are shown in Figure 4-50b. 

Now the after-inlet condition can be analyzed. 

Q„(I2) = 284,000 yd?/yr. (same as "before inlet") , 

%{2,3) = Q«(i,2) = 284,000 yd?/yr. (reach 2 is stable) . 

The gross transport rate across the inlet with the new y = 3.5 using 
Equation 4-24, is: 

Qn(2.5)(l+T) 

%{2,3) = (284,000 yd?/yr.) f^J , 
%{2,3) = 512,000 yd?/yr. . 
The inlet sink (Qp = 15 percent of (^(2,3) 

Q~2 = 512,000 yd.Vyr. X 0.15 , 
Q2 = 77,000 yd?/yr. 
The erosion value from Reach 3 now becomes : 

Reach 3 erosion = spit accretion + inlet sink - net littoral drift 
right of inlet. 

\3) «i3)) - %0,4) ^ ^2 - %i2,3) ' 

^(3) {^H3)) " 400,000 yd?/yr. + 77,000 yd?/yr. - 284,000 yd?/yr. , 

b(^) (q^(^^) = 193,000 yd?/yr. . 

4-138 



Nourishment needed to maintain historic erosion rate on Reach 3 beach 
is: 

Reach 3 nourishment = Reach 3 erosion "after inlet" - Reach 3 erosion 
"before inlet". 

^5(5) " \3) ('i'3i3)) after inlet - b^^^ (qj^^^) before inlet . 
^5(5) = 193,000 yd?/yr. - 82,000 yd?/yr. , 
Qj^^^ = 11 1,000 yd?/yr. . 

If reach 3 erosion is to be eliminated, it will be necessary to provide 
nourishment of 193,000 cubic yards per year. 

Q^, Qj^^, and Q^^ for the after-inlet condition are computed using 
Equation 4-24 and related equations. The after-inlet sand budget is 
shown in Figure 4-50c. 

************************************* 

4.8 ENGINEERING STUDY OF LITTORAL PROCESSES 

This section demonstrates the use of Chapter 4 in the engineering 
study of littoral processes. 

4.81 OFFICE STUDY 

The first step in the office phase of an engineering study of litto- 
ral processes is to define the problem in terms of littoral processes. 
The problem may consist of several parts, especially if the interests of 
local groups are in conflict. An ordering of the relative importance of 
the different parts may be necessary, and a complete solution may not be 
feasible. Usually, the problem will be stated in terms of the require- 
ments of the owner or local interests. For example, local interests may 
require a recreational beach in an area of limited sand, making it neces- 
sary to estimate the potential rates of longshore and onshore-offshore 
sand transport. Or a fishing community may desire a deeper channel in an 
inlet through a barrier island, making it necessary to study those litto- 
ral processes that will affect the stability and long-term navigability 
of the inlet, as well as the effect of the improved inlet on neighboring 
shores and the lagoon. 

4.811 Sources of Data . The next step is to collect pertinent data. If 
the problem area is located on a U.S. coastline, the National Shoreline 
Study may be consulted. This study can provide a general description of 
the area, and may give some indication of the littoral processes occur- 
ring in the vicinity of the problem area. 



4-139 



Historical records of shoreline changes are usually in the form of 
charts, surveyed profiles, dredging reports, beach replenishment reports 
and aerial photos. As an example of such historical data. Figure 4-52 
shows the positions of the shoreline at Sandy Hook, New Jersey, during 
six surveys from 1835 to 1932. Such shoreline changes are useful for 
computing longshore transport rates. The Corps of Engineers maintains, 
in its District and Division offices, survey, dredging, and other reports 
relating to Corps projects. Charts may be obtained from various federal 
agencies including the Defense Mapping Agency Hydrographic Center, Geolog- 
ical Survey, National Ocean Survey, and Defense Mapping Agency Topographic 
Center. A map called "Status of Aerial Photography," which may be obtained 
from the Map Information Office, Geological Survey, Washington, D.C. 20242, 
shows the locations and types of aerial photos available for the U.S., and 
lists the sources from which the photos may be requested. A description of 
a coastal imagery data bank can be found in the interim report by Szuwalski 
(1972). 

Other kinds of data usually available are wave, tide, and meteoro- 
logical data. Chapter 3 discusses wave and water level predictions; Sec- 
tion 4.3 discusses the effects of waves on the littoral zone; and Sub- 
section 4.34 presents methods of estimating wave climate and gives pos- 
sible sources of data. These referenced sections indicate the wave, tide, 
and storm data necessary to evaluate coastal engineering problems. 

Additional information can be obtained from local newspapers, court- 
house records, and from area residents. Local people can often identify 
factors that outsiders may not be aware of, and can also provide quali- 
titive information on previous coastal engineering efforts in the area 
and their effects. 

4.812 Interpretation of Shoreline Position . Preliminary interpretation 
of littoral processes is possible from the position of the shoreline on 
aerial photos and charts. Stafford (1971) describes a procedure for uti- 
lizing periodic aerial photographs to estimate coastal erosion. Used in 
conjunction with charts and topographic maps, this technique may provide 
quick and fairly accurate estimates of shoreline movement, although the 
results can be biased by the short-term effects of storms. 

Charts show the coastal exposure of a study site, and since exposure 
determines the possible directions from which waves reach the coast, expo- 
sure also determines the most likely direction of longshore transport. 

Direction of longshore transport may also be indicated by the posi- 
tion of sand accumulation and beach erosion around littoral barriers. A 
coastal structure in the surf zone may limit or prevent the movement of 
sand, and the buildup of sediment on one side of the littoral barrier serv- 
es as an indicator of the net direction of transport. This buildup can be 
determined from dredging or sand bypassing records or aerial photos. Fig- 
ure 4-53 shows the accumulation of sand on one side of a jetty. But wave 
direction and nearshore currents at the time of the photo indicate that 
transport then was in the opposite direction. Thus, an erroneous conclusion 



4-140 




Figure 4-52. Growth of Sandy Hook, New Jersey, 1835-1932 



4-141 



about the net transport might be made, if only wave patterns of this photo 
are analyzed. The possibility of seasonal or storm-induced reversals in 
sediment transport direction should be investigated by periodic inspections 
or aerial photos of the sand accumulation at groins and jetties. 




Figure 4-53. 



(4 December 1967) 
Transport Directions at New Buffalo Harbor Jetty on 
Lake Michigan 



The accumulation of sand on the updrift side of a headland is illus- 
trated by the beach north of Point Mugu in Figure 4-54. The tombolo in 
Figure 4-55 was created by deposition behind an offshore barrier (Grey- 
hound Rock, California). Where a beach is fixed at one end by a struc- 
ture or natural rock formation, the updrift shore tends to align perpen- 
dicular to the direction of dominant wave approach. (See Figures 4-55, 
and 4-56.) This alignment is less complete along shores with signifi- 
cant rates of longshore transport. 

Sand accumulation at barriers to longshore transport may also be 
used to identify nodal zones. There are two types of nodal zones: diver- 
gent and convergent. A divergent nodal zone is a segment of shore char- 
acterized by net longshore transport directed away from both ends of the 
zone. A convergent nodal zone is a segment of shore characterized by net 
longshore transport directed into both ends of the zone. 

Figure 4-56 shows a nodal zone of divergence centered around the 
fourth groin from the bridge on the south coast of Staten Island, Outer 
New York Harbor. Central Padre Island, Texas, is thought to be an example 



4-142 




(21 Moy 1972; 



Figure 4-54. Sand Accumulation at Point Mugu, California 



4-143 




(29 August 1972) 



Figure 4-55. Tombolo and Pocket Beach at Greyhound 
Rock, California 



4-144 







' T 


■ jr' 


* -7 


-%:«#"' 


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Zone ,,-'•*' T«Ow 








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( 14 September 1969) 



F:.gure 4-56. A Nodal Zone of Divergence Illustrated 
by Sand Accumulation at Groins, South 
Shore, Staten Island, New York 



4-145 



of a convergent nodal zone. (Watson, 1971.) Nodal zones o£ divergence 
are more common than nodal zones of convergence, because longshore trans- 
port commonly diverges at exposed shores and converges toward major gaps 
in the ocean shore, such as the openings of New York Harbor, Delaware Bay, 
and Chesapeake Bay. 

Nodal zones are usually defined by long-term average transport direc- 
tions, but because of insufficient data, the location of the mid-point of 
the nodal zone may be uncertain by up to 10' s of miles. In addition, the 
short-term nodal zone most probably shifts along the coast with changes 
in wave climate. 

The existence, location, and planform of inlets can be used to inter- 
pret the littoral processes of the region. Inlets occur where tidal flow 
is sufficient to maintain the openings against longshore transport which 
acts to close them. (e.g., Bruun and Gerritsen, 1959.) The size of the 
inlet opening depends on the tidal prism available to maintain it. (O'Brien, 
1969.) The dependence of inlet size on tidal prism is illustrated by Fig- 
ure 4-57, which shows three bodies of water bordering the beach on the south 
shore of Long Island, New York. The smallest of these (Sagaponack Pond) 
is sealed off by longshore transport; the middle one (Mecox Bay) is partly 
open; and the largest (Shinnecock Bay) is connected to the sea by Shinne- 
cock Inlet, which is navigable. 




( 14 September 19691 



Figure 4-57. South Shore of Long Island, N.Y., Showing Closed, 
Partially Closed, and Open Inlets 



4-146 



Detailed study of inlets through barrier islands on the U.S. Atlan- 
tic and gulf coasts shows that the shape of the shoreline at an inlet can 
be classified in one of four characteristic planforms. (See Figure 4-58, 
adapted from Galvin, 1971,) Inlets with overlapping offset (Fig. 4-59) 
occur only where waves from the updrift side dominate longshore transport. 
Wliere waves from the updrift side are less dominant, the updrift offset 
(Fig. 4-59) is common. Where waves approach equally from both sides, in- 
lets typically have negligible offset. (See Figure 4-60.) IVhere the sup- 
ply of littoral drift on the updrift side is limited, and the coast is 
fairly well exposed, a noticeable downdrift offset is common as, for exam- 
ple, in southern New Jersey and southern Delmarva. (See Hayes, et al., 
(1970.) These planform relations to littoral processes have been found 
for inlets through sandy barrier islands, but they do not necessarily hold 
at inlets with rocky boundaries. The relations hold regionally, but tem- 
porary local departures due to inlet migration may occur. 

4.82 FIELD STUDY 

A field study of the problem area is usually necessary to obtain 
types of data not found in the office study, to supplement incomplete 
data, and to serve as a check on the preliminary interpretation and corre- 
lations made from the office data. Information on coastal processes may 
be obtained from wave gage data and visual observations, sediment sam- 
pling, topographic and bathymetric surveys, tracer programs, and effects 
of natural and manmade structures. 

4.821 Wave Data Collection . A wave-gaging program yields height and 
period data. However, visual observations may currently be the best 
source of breaker direction data. Thompson and Harris (1972) deter- 
mined that 1 year of wave-gage records provides a reliable estimate of 
the wave height frequency distribution. It is reasonable to assume that 
the same is true for wave direction. 

A visual observation program is inexpensive, and may be used for 
breaker direction and for regional coverage when few wave-gage records 
are available. The observer should be provided with instructions, so 
that all data collected will be uniform, and contact between observer 
and engineer should be maintained. 

4.822 Sediment Sampling . Sediment sampling programs are described in 
Section 4.26. Samples are usually surface samples taken along a line 
perpendicular to the shoreline. These are supplemented by borings or 
cores as necessary. Complete and permanent identification of the sam- 
ple is important. 

4.823 Surveys . Most engineering studies of littoral processes require 
surveying the beach and nearshore slope. Successive surveys provide 
data on changes in the beach due to storms, or long-term erosion or 
accretion. If beach length is also considered, an approximate volume 
of sand eroded or accreted can be obtained which provides information 
for the sediment budget of the beach. The envelope of a profile defines 

4-147 



Qrt 



Q/t 




Overlapping Offset 



Qrt 



Adequate 
updrift source j^ 



Q/t 



Length of arrows indicotes 
relative magnitude of 
Longshore Tronsport Rate 



Updrift Offset 




Inadequate 
updrift source 



Downdrift Offset 




Qrt Qit 



Negligible Offset 




(Galvin,l97l ) 



Figure 4-58. Four Types of Barrier Island Offset 



4-148 




(14 September 1969) 
Figure 4-59. Fire Island Inlet, New York - Overlapping Offset 



BAY 




x^ 






• 


^j|||/ n ANTIC 







( 13 Morch 19631 
Figure 4-60. Old Drum Inlet, North Carolina - Negligible Offset 



4-149 



fluctuations of sand level at a site (Everts, 1973), and thus provides 
data useful in beach fill and groin design. 

Methods for obtaining beach and nearshore profiles, and the accu- 
racy of the resulting profiles are discussed in Section 4.514. 

4.824 Tracers . It is often possible to obtain evidence on the direction 
of sediment movement and the origins of sediment deposits by the use of 
tracer materials which move with the sediment. Fluorescent tracers were 
used to study sand migration in and around South Lake Worth Inlet, Florida. 
(Stuiver and Purpura, 1968.) Radioactive sediment tracer tests were con- 
ducted to determine whether potential shoaling material passes through or 
around the north and south jetties of Galveston Harbor. (Ingram, Cummins, 
and Simmons, 1965.) 

Tracers are particles which react to fluid forces in the same manner 
as particles in the sediment whose motion is being traced, yet which are 
physically identifiable when mixed with this sediment. Ideally, tracers 
must have the same size distribution, density, shape, surface chemistry, 
and strength as the surrounding sediment, and in addition have a physical 
property that easily distinguishes them from their neighbors. 

Three physical properties have been used to distinguish tracers: 
radioactivity, color, and composition. Tracers may be either naturally 
present or introduced by man. There is considerable literature on recent 
investigations using or evaluating tracers including reviews and biblio- 
graphy: (Duane and Judge, 1969; Bruun, 1966; Galvin, 1964; and Huston, 
1963); models of tracer motion: (James, 1970; Galvin, 1964; Hubbell and 
Sayre, 1965; and Duane, 1970); and use in engineering problems: (Hart, 
1969; Cherry, 1965; Cummins, 1964; and Duane, 1970). 

a. Natural Tracers . Natural tracers are used primarily for back- 
ground information about sediment origin and transport directions, i.e., 
for studies which involve an understanding of sediment patterns over a 
long period of time. 

Studies using stable, nonradioactive natural tracers may be based 
on the presence or absence of a unique mineral species, the relative 
abundance of a particular group of minerals within a series of samples, 
or the relative abundance and ratios of many mineral types in a series 
of samples. Although the last technique is the most complex, it is often 
used, because of the large variety of mineral types normally present in 
sediments and the usual absence of singularly unique grains. The most 
suitable natural tracers are grains of a specific rock type originating 
from a localized specific area. 

Occasionally, characteristics other than mineralogy are useful for 
deducing source and movement patterns. Krinsley, et al., (1964) devel- 
oped a technique for the study of surface textures of sand grains with 
electron microscopy, and applied the technique to the study of sand trans- 
port along the Atlantic shore of Long Island. Naturally occurring radio- 
active materials in beach sands have also been used as tracers. (Kamel, 
1962.) 

4-150 



One advantage of natural tracers is their tendency to "average" out 
short-term trends and provide qualitatively accurate historical background 
information on transport. Their use requires a minimum amount of field 
work and a minimum number of technical personnel. Disadvantages include 
the irregularity of their occurrence, the difficulty in distinguishing the 
tracer from the sediment itself, and a lack of quantitative control on 
rates of injection. In addition, natural tracers are unable to reveal 
short-term changes in the direction of transport and changes in material 
sources. 

Judge (1970) found that heavy mineral studies were unsatisfactory as 
indicators of the direction of longshore transport for beaches between 
Point Conception and Ventura, California, because of the lack of unique 
mineral species and the lack of distinct longshore trends which could be 
used to identify source areas. North of Point Conception, grain size and 
heavy mineral distribution indicated a net southward movement. Cherry 
(1965) concluded that the use of heavy minerals as an indicator of the 
direction of coastal sand movement north of Drakes Bay, California was 
generally successful. 

b. Artificial Tracers . Artificial tracers may be grouped into two 
general categories: radioactive or nonradioactive. In either case, the 
tracers represent particles that are placed in an environment selected 
for study, and are used for relatively short-term studies of sediment 
dispersion. 

While particular experiments employ specific sampling methods and 
operational characteristics, there are basic elements in all tracing 
studies. These are: selection of a suitable tracer material, tagging 
the particle, placing the particle in the environment, and detection of 
the particle. 

Colored glass, brick fragments and oolitic grains are a few examples 
of nonradioactive particles that have been used as tracers. The most com- 
monly used stable tracer is made by coating indigenous grains with bright 
colored paint or flourescent dye. (Yasso, 1962; Ingle, 1966; Stuvier and 
Purpura, 1968; Kidson and Carr, 1962; and Teleki, 1966.) The dyes make 
the grains readily distinguishable among large sample quantities, but do 
not significantly alter the physical properties of the grains. The dyes 
must be durable enough to withstand short-term abrasion. The use of paints 
and dyes as tracer materials offers advantages over radioactive methods in 
that they require less sophisticated equipment to tag and detect the grains, 
and do not require licensing or the same degree of safety precautions. How- 
ever, less information is obtained for the same costs, and generally in a 
less timely matter. 

When using nonradioactive tracers, samples must be collected and 
removed from the environment to be analyzed later by physically counting 
the grains. For fluorescent dyes and paints, the collected samples are 
viewed under an ultraviolet lamp and the coated grains counted. 

For radioactive tracer methods, the tracer may be radioactive at the 
time of injection or it may be a stable isotope capable of being detected 
by activation after sampling. The tracer in the grains may be introduced 

4-151 



by a number of methods. Radioactive material has been placed in holes 
drilled in a large pebble. It has been incorporated in molten glass 
which, when hardened, is crushed and resized (Sato, et al., 1962; and 
Taney, 1963). Radioactive material has been plated on the surface of 
natural sediments. (Stephens, et al., 1968.) Radioactive gas (krypton 
85 and xenon 133) has been absorbed into quartz sand. (Chleck, et al., 
1963; and Acree, et al., 1969.) 

In 1966, the Coastal Engineering Research Center, in cooperation 
with the Atomic Energy Commission, initiated a multiagency program to 
create a workable radioisotopic sand tracing (RIST) program, for use in 
the littoral zone. (Duane and Judge, 1969.) Tagging procedures (by 
surface plating with gold 198-199), instrumentation, field surveys and 
data handling techniques were developed which permit collection and 
analysis of over 12,000 bits of information per hour over a survey track 
about 18,000 feet long. 

These recent developments in radioactive tracing permit in situ 
observations and faster data collection over much larger areas (Duane, 
1970) than has been possible using fluorescent or stable isotope tracers. 
However, operational and equipment costs of radionuclide tracer programs 
can be high. 

Accurate determination of long-term sediment transport volume is not 
yet possible from a tracer study, but qualitative data on sediment move- 
ment useful for engineering purposes can be obtained. 

Experience has shown that tracer tests can give information on 
direction of movement, dispersion, shoaling sources, relative velocity 
and movement in various areas of the littoral zone, means of natural by- 
passing, and structure efficiency. Reasonably quantitative data on move- 
ment or shoaling rates can be obtained for short-time intervals. It 
should be emphasized that this type of information must be interpreted 
with care, since the data are generally determined by short-term littoral 
transport phenomena. However, tracer studies conducted repeatedly over 
several years at the same location could result in estimates of longer 
term littoral transport. 

4.83 SEDIMENT TRANSPORT CALCULATIONS 

4.831 Longshore Transport Rate . The example calculation of a sediment 
budget in Section 4.76 is typical in that the magnitude of the longshore 
transport rate exceeds by a considerable margin any other element in the 
budget. For this reason, it is essential to have a good estimate of the 
longshore transport rate in an engineering study of littoral processes. 

A complete description of the longshore transport rate requires 
knowledge of two of the five variables 

(Q,,,Qe,,Q^,Q„,7). 

defined by Equations 4-21, 4-22, and 4-23. If any two are known, the 
remaining three can be obtained from the three equations. 



4-152 



Section 4.531 describes four methods for estimating longshore trans- 
port rate, and Sections 4.532 through 4.535 describe in detail how to use 
two of these four methods. (See Methods 3 and 4.) 

One approach to estimating longshore transport rate is to adopt a 
proven estimate from a nearby locality, after making allowances for local 
conditions. (See Method 1.) It requires considerable engineering judge- 
ment to determine whether the rate given for the nearby locality is a 
reliable estimate, and, if reliable, how the rate needs to be adjusted 
to meet the changed conditions at the new locality. 

Method 2 is an analysis of historical data. Such data may be found 
in charts, maps, aerial photography, dredging records, beach fill records, 
and related information. Section 4.811 describes some of these sources. 

To apply Method 2, it is necessary to know or assume the transport 
rate across one end of the littoral zone being considered. The most suc- 
cessful applications of Method 2 have been where the littoral zone is 
bounded on one end by a littoral barrier which is assumed to completely 
block all longshore transport. The existence of such a complete littoral 
barrier implies that the longshore transport rate is zero across the 
barrier, and this satisfies the requirement that the rate be known across 
the end of the littoral zone being considered. Examples of complete lit- 
toral barriers include large jetties immediately after construction, or 
spits building into deep quiet water. 

Data on shoreline changes permit estimates of rates of erosion and 
accretion that may give limits to the longshore transport rate. Figure 
4-50 is a shoreline change map which was used to obtain the rate of 
transport at Sandy Hook, New Jersey. (Caldwell, 1966.) 

Method 3 (the energy flux method) is described in Section 4.532 
with a worked example in Section 4.533. Method 4 (the empirical pre- 
diction of gross longshore transport rate) is described in Section 4.534 
with a worked example in Section 4.535. The essential factor in Methods 
3 and 4, and often in Method 1, is the availability of wave data. Wave 
data applicable to studies of littoral processes are discussed in detail 
in Section 4.3. 

4.832 Onshore-Offshore Motion . Typical problems requiring knowledge of 
onshore-offshore sediment transport are described in Section 4.511. Four 
classes of problems are treated: 

(1) The seaward limit of significant sediment transport. Avail- 
able field data and theory suggest that waves are able to move sand dur- 
ing some days of the year over most of the Continental Shelf. However, 
field evidence from bathymetry and sediment size distribution suggest 
that the zone of significant sediment transport is confined close to 
shore where bathymetric contours approximately parallel the shoreline. 
The depth to the deepest shore-parallel contour tends to increase with 
average wave height, and typically varies from 15 to 60 feet. 



4-153 



(2) Sediment transport in the nearshore zone. Seaward of the 
breakers, sand is set in motion by waves moving over ripples, either 
rolling the sand as bed load, or carrying it up in vortices as suspended 
load. The sand, once in motion, is transported by mean tidal and wind- 
induced currents and by the mass transport velocity due to waves. The 
magnitude and direction of the resulting sediment transport are uncer- 
tain under normal circumstances, although mass transport due to waves 

is more than adequate to return sand lost from the beach during storms. 
It appears that bottom mass transport acts to keep the sand close to the 
shore, but that some material, probably finer sand, escapes offshore as 
the result of the combined wind- and wave-induced bottom currents. 

(3) The shape and expectable changes in shape of nearshore and 
beach profiles. Storms erode beaches to produce a simple concave-up 
beach profile with deposition of the eroded material offshore. Rates of 
erosion due to individual storms vary from a few cubic yards per foot to 
10 's of cubic yards per foot of beach front. The destructiveness of the 
storm in producing erosion depends on its intensity, duration, and orienta- 
tion, especially as these factors affect the elevation of storm surge and 
the wave height and direction. Immediately after a storm, waves begin to 
return sediment to the eroded beach, either through the motion of bar-and- 
trough (ridge-and-runnel) systems, or by berm building. The parameter, 

Fo = Ho/iVfT), given by Equation 4-20 determines whether the beach erodes 
or accretes under given conditions. If V^ is above critical value 
between 1 and 2 the beach erodes. (See Figures 4-29, and 4-30.) 

(4) The slope of the foreshore. There is a tendency for the fore- 
shore to become steeper as grain size increases, and to become flatter 

as mean wave height increases. Data for this relation exhibit much scat- 
ter and quantitative relationships are difficult to predict. 

4.833 Sediment Budget. Section 4.76 siommarizes material on the sediment 
budget. Table 4-14 tabulates the elements of the sediment budget and in- 
dicates the importance of each element. Table 4-13 classifies the ele- 
ments of the sediment budget. 

A sediment budget carefully defines the littoral control volume, 
identifies all elements transferring sediment to or from the littoral 
control volume, ranks the elements by their magnitude, and provides an 
estimate of unknown rates by the balancing of additions against losses 
(Equation 4-46) . 

If prepared with sufficient data and experience, the budget permits 
an estimate of how proposed improvements will affect neighboring segments 
of the littoral zone. 



4-154 



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ACREE, E.H., et al., "Radioisotopic S'lnd Tracer Study (RIST) , Status 
Report for May 1966-April 1968," OKNL-4341, Contract No. W-7405-eng- 
26, Oak Ridge National Laboratory (operated by Union Carbide Corp., 
for the U.S. Atomic Energy Commission), 1969. 

AKYUREK, M. , "Sediment Suspension by Wave Action Over a Horizontal Bed," 
Unpublished Thesis, University of Iowa, Iowa City, Iowa, July 1972. 

ARTHUR, R.S., "A Note on the Dynamics of Rip Currents," Journal of 
Geophysioal Researah, Vol. 67, No. 7, July 1962. 

AYERTON, H, "The Origin and Growth of Ripple Marks," Philosophical 
Transaotions of the Royal Society of London, Ser. A, Vol. 84, 1910, 
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BAGNOLD, R„A„, "Mechanics of Marine Sedimentation," The Sea, Vol. 3, 
Wiley, New York, 1963, pp. 507-528. 

BAJORUNAS, L„, and DUANE, D.B., "Shifting Offshore Bars and Harbor Shoal- 
ing," Journal of Geophysioal Research, Vol. 72, 1967, pp. 6195-6205. 

BASCOM, W.N., "The Relationship Between Sand Size and Beach-Face Slope," 
Transactions of the American Geophysical Union, Vol. 32, No. 6, 1951. 

BASCOM, W.N. , Manual of Amphibious Oceanography, Vol. 2, Sec. VI, Univer- 
sity of California, Berkeley, Calif., 1952. 

BASCOM, W.N., "Waves and Beaches," Beaches, Ch. IX, Doubleday, New York, 
1964, pp. 184-212. 

BASH, B.F., "Project Transition; 26 April to 1 June 1972," Unpublished 
Research Notes, U.S. Army, Corps of Engineers, Coastal Engineering 
Research Center, Washington, D.C., 1972. 

BERG, D.W., "Systematic Collection of Beach Data," Proceedings of the 
11th Conference on Coastal Engineering, London, Sept. 1968. 

BERG, D.W., and DUANE, D.B., "Effect of Particle Size and Distribution on 
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April 1968. 

BERTMAN, D„Y., SHUYSKIY, Y.D., and SHKARUPO I.V., "Experimental Study 
of Beach Dynamics as a Function of the Prevailing Wind Direction and 
Speed," Oceanology, Translation of the Russian Journal, Okeanologiia 
(Academy of Sciences of the U.S.S.R.), Vol. 12, Feb. 1972. 

BIGELOW, H.B., and EDMONDSON, W.T., "Wind Waves at Sea Breakers and Surf," 
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4-155 



BIJKER, E.W., "Bed Roughness Influence on Computation of Littoral Drift," 

Abstracts of the 12th Coastal Engineering Conference^ Washington, D.C., 
1970. 

BODINE, B.R., "Hurricane Surge Frequency Estimated for the Gulf Coast of 
Texas," Bl-26, U.S. Army, Corps of Engineers, Coastal Engineering 
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BOWEN, A.J., and INMAN, D.L., "Budget of Littoral Sands in the Vicinity 
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BOWMAN, R.S., "Sedimentary Processes Along Lake Erie Shore: Sandusky Bay, 
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BRETSCHNEIDER, C.L., "Fundamentals of Ocean Engineering - Part 1 - 
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BRETSCHNEIDER, C.L., "Field Investigations of Wave Energy Loss in 
Shallow Water Ocean Waves," TM-46, U.S. Army, Corps of Engineers, 
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BRETSCHNEIDER, C.L., and REID, R.O., "Modification of Wave Height Due 
to Bottom Friction, Percolation, and Refraction," TM-45, U.S. Army, 
Corps of Engineers, Beach Erosion Board, Washington, D.C, Oct. 1954. 

BRICKER, O.P., ed.. Carbonate Sediments, No. 19, The Johns Hopkins 
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BROWN, Ed., "Inlets on Sandy Coasts," Proceedings of the American 
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BRUUN, P., "Sea-Level Rise as a Cause of Shore Erosion," Journal of the 
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117-130. 

BRUUN, P., "Longshore Currents and Longshore Troughs," Journal of Geo- 
physical Research, Vol. 68, 1963, pp. 1065-1078. 

BRUUN, P., "Use of Tracers in Coastal Engineering," Shore and Beach, 
No. 34, pp. 13-17, Nuclear Science Abstract, 20:33541, 1966. 

BRUUN, P„ and GERRITSEN, F., "Natural Bypassing of Sand at Inlets," 

Joujpnal of the Waterways and Harbors Division, ASCE, Dec. 1959. 

BUMPUS, D.F., "Residual Drift Along the Bottom on the Continental Shelf 
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4-156 



BYRNE, R„J., "Field Occurrences of Induced Multiple Gravity Waves," 

Journal of Geophysical Besearoh^ Vol. 74, No. 10, May 1969, pp. 
2590-2596. 

CALDWELL, J.M., "Wave Action and Sand Movement Near Anaheim Bay, 
California," TM-68, U.S. Army, Corps of Engineers, Beach Erosion 
Board, Washington, D.C., Feb. 1956. 

CALDWELL, J„M., "Shore Erosion by Storm Waves," MP 1-59, U.S. Army, 
Corps of Engineers, Beach Erosion Board, Washington, D.C., Apr. 
1959. 

CALDWELL, J„M., "Coastal Processes and Beach Erosion," Journal of the 
Boston Society of Civil Engineers^ Vol. 53, No. 2, Apr. 1966, pp. 
142-157, 

CARSTENS, M.K., NEILSON, M. , and ALTINBILEK, H.D., "Bed Forms Generated 
in the Laboratory Under an Oscillatory Flow: Analytical and Experi- 
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ing Research Center, Washington, D.C. , June 1969. 

CARTWRIGHT, D.E., "A Comparison of Instrumental and Visually Estimated 
Wave Heights and Periods Recorded on Ocean Weather Ships," National 
Institute of Oceanography, Oct. 1972. 

CASTANHO, J., "Influence of Grain Size on Littoral Drift," Abstracts of 
the 12th Coastal Engineering Conference, Washington, D.C, 1970, 

CHERRY, Jo, "Sand Movement along a Portion of the Northern California 
Coast," TM-14, U.S. Army, Corps of Engineers, Coastal Engineering 
Research Center, Oct. 1965. 

CHLECK, D., et al., "Radioactive Kryptomates," International Journal of 
Applied Radiation and Isotopes, Vol. 14, 1963, pp. 581-610. 

CHOW, V,T., ed.. Handbook of Applied Hydrology, McGraw-Hill, New York, 
1964. 

COLBY, B.C., and CHRISTENSEN, R.P., "Visual Accumulation Tube for Size 
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COLONY, R„J., "Source of the Sands on South Shore of Long Island and the 
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pp. 150-159. 

COOK, D.O., "Sand Transport by Shoaling Waves," Ph.D. Thesis, University 
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4-157 



COOPER, W.S., "Coastal Sand Dunes of Oregon and Washington," Memoir No. 
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Samples," Report No. 4, St. Paul U.S. Engineer District Sub-Office, 
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CRICKMORE, J.M., and WATERS, C.B., "The Measurement of Offshore Shingle 
Movement," 23th International Conferenoe on Coastal Engineering, 
Vancouver, B.C., Canada, July 1972. 

CUMMINS, R.S., Jr., "Radioactive Sediment Tracer Tests, Cape Fear River, 
North Carolina," MP No. 2-649, Waterways Experiment Station, U.S. 
Army, Corps of Engineers, Vicksburg, Miss., May 1964. 

CURRAY, J.R., "Late Quartemary History, Continental Shelves of the United 
States," The Quartemary of the U.S., Princeton University Press, 
Princeton, N.J., 1965, pp. 723-735. 

DABOLL, J.M., "Holocene Sediments of the Parker River Estuary, Massa- 
chusetts," Contribution No. 3-CRG, Department of Geology, University 
of Massachusetts, June 1969. 

DARLING, J.M., "Surf Observations Along the United States Coasts," Journal 
of the Waterways and Harbors Division, ASCE, WWl, Feb. 1968, pp. 11-21. 

DARLING, J.M., and DUMM, D.G., "The Wave Record Program at CERC," MP 
1-67, U.S. Army, Corps of Engineers, Coastal Engineering Research 
Center, Washington, D.C., Jan. 1967. 

DAS, M.M., "Longshore Sediment Transport Rates: A Compilation of Data," 
MP 1-71, U.S. Army, Corps of Engineers, Coastal Engineering Research 
Center, Washington, D.C., Sept. 1971. 

DAS, M.M., "Suspended Sediment and Longshore Sediment Transport Data 
Review," IZth International Conference on Coastal Engineering, 
Vancouver, B.C., Canada, July 1972. 

DAVIS, R.A., Jr., "Sedimentation in the Nearshore Environment, South- 
eastern Lake Michigan," Thesis, University of Illinois, Urbana, 
Illinois, 1964. 

DAVIS, R.A., Jr., "Beach Changes on the Central Texas Coast Associated 
with Hurricane Fern, September 1971," Vol. 16, Contributions in Marine 
Science, University of Texas, Marine Science Institute, Port Aransas, 
Tex., 1972. 

DAVIS, R.A., Jr., and FOX, W.T., "Beach and Nearshore Dynamics in Eastern 
Lake Michigan," TR No. 4, ONR Task No. 388-092/10-18-68, (414), Office 
of Naval Research, Washington, D.C., June 1971. 

DAVIS, R.A., Jr., and FOX, W.T., "Coastal Processes and Nearshore Sand 
Bars," Journal of Sedimentary Petrology, Vol. 42, No. 2, June 1972, 
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4-158 



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